6.1 Add & Subtract Polynomial Expression & Functions

Save this PDF as:

Size: px
Start display at page:

Download "6.1 Add & Subtract Polynomial Expression & Functions"

Transcription

1 6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic funciton, vertex, and cubic function. 2. Add & Subtract Polynomials 3. Evaluate polynomial funcitons 4. Know typical graphs of polynomial funcitons. 5. Find the sum function and difference funciton. 6. Use the sum and difference functions to model situations.

2 6.1 Add & Subtract Polynomials (Page 2 of 33) Polynomials A term is a constant, a variable, or a product of a constant and one or more variables raised to powers. A monomial is a constant, a variable, or a product of a constant and one or more variables raised to counting number (i.e. positive integer) powers. A binomial is the sum of two monomials. e.g. 45,!7x!x 1/2, 2x 3 y!4 e.g. 5, 3x x, 2x 3 y 4 e.g. 3x + 2y A trinomial is the sum of three monomials. e.g. x 2! x + 7 A polynomial is a monomial or a sum of monomials. e.g. 5x 3! 2x 2 + 7x! 9, 4x 5 y! xy 2,!2x + 8, 3, x The polynomial 4x 3! 2x 2 + x +12 is a polynomial in one variable with four terms, namely 4x 3,!2x 2, x and 12. By convention, we write polynomial so that the exponents decrease from left to right, which is called descending order. The degree of a term in one variable is the exponent on the variable. For example, the degree of 2y 4 is four. The degree of a term in two or more variables is the sum of the exponents on the variables. For example, the degree of!x 3 y 4 is seven. The degree of a polynomial is the highest degree of any term in the polynomial. The coefficient of a term is the constant factor of the term. The leading coefficient of a polynomial is the coefficient of the term with the highest degree. A linear polynomial has degree 1. A quadratic polynomial has degree 2. A cubic polynomial has degree 3. A constant has degree 0.

3 6.1 Add & Subtract Polynomials (Page 3 of 33) Example 1 Write each polynomial in descending order. Then use words such as linear, quadratic, cubic, polynomial, degree, one variable, two variables, coefficient and leading coefficient to describe each term and each expression. a.!3+ 8x! 4x 2 b. 4x + 2x c. 12 d. 3a 6 b 2 + 7ab 3! 3a 4 b Like Terms Like terms are either constant terms, or terms with identical variable parts (including exponents). The distributive property makes it possible to combine like terms by adding the coefficients while leaving the variable parts unchanged. For example,!3x 2 + 7x 2 = (!3+ 7)x 2 = 4x 2 Example 2 Simplify (combine like terms in descending order). 1. 5a 3! 4a 2 + 7a 3! a p 2 t 2 + p 2 t! 8 p 3 t 2! 9 p 2 t

4 6.1 Add & Subtract Polynomials (Page 4 of 33) Example 3 Perform the indicated operation and simplify. 1. (8a 2! 7ab + 2b 2 ) + (3a 2 + 4ab! 7b 2 ) 2. (5x 3! x + 7)! (!8x 3 + 3x 2! 6) Polynomial Function A polynomial function is a function that can be written in the form f (x) = polynomial. Some examples are f (x) = 4x 5! 3x 3 + 7, g(x) = 2.3x 3! 7x, h(x) = 7 9 x. A polynomial function of degree two is called a quadratic function. Quadratic Function in Standard Form A quadratic function in standard form is a function that can be written as f (x) = ax 2 + bx + c, where a! 0. Furthermore, the quadratic term is ax 2, the linear term is bx, and the constant term is c. For example, f (x) =!x 2 + 7x is quadratic because a =!1, b = 7 and c = 0. Example 4 For f (x) =!2x 2 + 5x!1, find each of the following. 1. f (4) 2. f (!3) 3. f (0)

5 6.1 Add & Subtract Polynomials (Page 5 of 33) Example 5 Sketch the graph of f (x) = x 2 the TABLE and graphing capabilities of your calculators.. 5 y x f (x) = x x Parabolas The graph of a quadratic function is called a parabola and has the shape illustrated. Parabolas can open downward like function g, or open upward like function f. If the parabola opens upward, then the vertex is located at the minimum point of the graph. If the parabola opens downward, then the vertex is located at the maximum point of the graph. The vertical line that goes through the vertex is called the axis of symmetry of the parabola.

6 6.1 Add & Subtract Polynomials (Page 6 of 33) Example 6 Reading Parabolas 1. Identify the vertex in the graph of function f. Is it a maximum point or minimum point? What is the equation of the axis of symmetry? 2. Identify the vertex in the graph function g. Is it a maximum point or minimum point? What is the equation of the axis of symmetry? 3. Find f (6) 4. Find x when f (x) = 4 5. Find g(!5) 6. Find x when g(x) =!8

7 6.1 Add & Subtract Polynomials (Page 7 of 33) Cubic Function A cubic function is a 3 rd -degree polynomial function and can be written in the form f (x) = ax 3 + bx 2 + cx + d, where a! 0. Example 7 Cubic Function Sketch the graph of f (x) = x 3. y x f (x) = x x Graphs of Typical Cubic Functions See note-guide, p. 33 for more information. y =!0.5(x! 3) 3 y = 0.25(x + 6)(x! 3) 2 y = (x + 2)(x! 2)(x! 5)

8 6.1 Add & Subtract Polynomials (Page 8 of 33) Sum and Difference Functions If f and g are functions and x is in the domain of both functions, then we can for the following functions: 1. Sum Function f + g ( f + g)(x) = f (x) + g(x) 2. Difference Function f - g ( f! g)(x) = f (x)! g(x) Example 8 Find the Sum & Difference Functions Let f (x) = 5x 2! x + 3 and g(x) =!2x 2 + 9x! Find the equation for f + g. 2. Find ( f + g)(2) 3. Find the equation for f g. 4. Find ( f! g)(2)

9 6.1 Add & Subtract Polynomials (Page 9 of 33) Example 9 Modeling The enrollments (in millions) at U.S. colleges W (t) and M (t) for women and men, respectively, are modeled by the system W (t) = 0.15t M (t) = 0.072t where t is the number of years since Find the equation for the sum function W + M. 2. Find (W + M )(31) and explain its meaning in this situation. 3. Find the equation for the difference function W - M. 4. Find (W! M )(31) and explain its meaning in this situation.

10 6.2 Multiplying Polynomial Expressions & Functions (Page 10 of 33) 6.2 Multiplying Polynomial Expressions & Functions The Factored form of the expression is written as a product. Multiplying a(b+c) = ab+ ac Factoring After multiplying, the expression is written as a sum. Example 1 Find the product (multiply). 1. 4x(x! 6) 2.!2a(3! 5a) Example 2 Finding Products Find the product (multiply). 1. (c + 4)(c + 5) 2. (x! 7)(x + 4) 3. (5 p! 6w)(3p + 2w) 4. (2a 2! 5b 2 )(4a 2! 3b 2 )

11 6.2 Multiplying Polynomial Expressions & Functions (Page 11 of 33) Example 3 Find the product (multiply). 1. 4x(x 2 + 2)(x! 3) 2. (2x + y)(5x 2! 3xy + 4y 2 ) 3. (x 2! 3x + 2)(3x 2 + x! 5) Example 4 Square a Binomial Find the product (multiply). 1. (b! 6) 2 2. (2x + 7) 2 Squaring a Binomial 1. (a + b) 2 = a 2 + 2ab+ b 2 2. (a! b) 2 = a 2! 2ab+ b 2 (a + b) 2 (a! b) 2 = First binomial term squared + Twice the product of the two binomial terms + Second binomial term squared

12 6.2 Multiplying Polynomial Expressions & Functions (Page 12 of 33) Squaring a Binomial 1. (a + b) 2 = a 2 + 2ab+ b 2 2. (a! b) 2 = a 2! 2ab+ b 2 (a + b) 2 (a! b) 2 = First binomial term squared + Twice the product of the two binomial terms + Second binomial term squared Example 5 1. Expand ( y + 7) 2 2. Expand (x + 5) 2 3. Expand (2b! 7) 2 4. Expand (5! 2x) 2 5. Expand (3a + 4b) 2 6. Expand (!5x + 7 y) 2

13 6.2 Multiplying Polynomial Expressions & Functions (Page 13 of 33) Product of Binomial Conjugates The binomials 2x! 7 and 2x + 7 are binomial conjugates. In general, the sum and difference of two terms ( A + B and A! B) are binomial conjugates of each other. Example 6 Find the product. 1. (c + 6)(c! 6) The Product of Binomial Conjugates = Difference of Two Squares (a + b)(a! b) = a 2! b 2 2. (x! 5)(x + 5) 3. (2b! 7)(2b + 7) 4. (6! 5x)(6 + 5x) 5. (4m 2! 7rt)(4m 2 + 7rt) 6. (x + 3)(x! 3)(x 2 + 9)

14 6.2 Multiplying Polynomial Expressions & Functions (Page 14 of 33) Example 7 For f (x) = x 2! 5x, find the following. 1. f (a! 3) 2. f (a + 2)! f (a) Example 8 Write f (x) =!3(x! 4) in standard form ( f (x) = ax 2 + bx + c ). Product Function If f and g are functions and x is in the domain of both functions, then we can form the product function f! g : ( f! g)(x) = f (x)! g(x) Example 9 Let f (x) =!3x + 7 and g(x) = 5x! 2. Find 1. ( f! g)(x) 2. ( f! g)(2)

15 6.2 Multiplying Polynomial Expressions & Functions (Page 15 of 33) Example 10 Let C(t) = 2.75t +102 represent the annual cost (in dollars) of prisons per person in the U.S. for t years since Let P(t) = 3.3t represent the U.S. population (in millions of people) at t years since Find the equation for the product function C! P. 2. Perform unit analysis on the function C(t)! P(t). That is what are the units of the product function. 3. Find (C! P)(20). Explain its meaning in this application. 4. Use a graphing calculator to determine whether the function C! P is increasing, decreasing, or neither for values of t between 5 and 20. What does the result mean in this situation?

16 6.3 Factoring Quadratic Polynomials (Page 16 of 33) 6.3 Factoring x 2 + bx + c = 1x 2 + bx + c Notice the patterns that develops in the following products. Last terms in the factored form The coefficient on the quadratic term is one (i.e. x 2 =1x 2 ) The coefficient of x in expanded form is the sum of the last terms in factored form (x + p)(x + q) = x 2 + qx + px + pq = x 2 + ( p + q)x + pq (x + 3)(x + 4) = x x + 3x +12 = x x +12 (x! 4)(x! 6) = x 2! 6x! 4x + 24 = x 2!10x + 24 (x + 5)(x! 7) = x 2! 7 x + 5x! 35 = x 2! 2x! 35 (x! 3)(x + 9) = x x! 3x! 27 = x x! 27 The constant term in expanded form is the product of the last terms in factored form Three Observations 1. The coefficient on the quadratic term is one (i.e. x 2 = 1x 2 ). 2. The constant term in each trinomial (i.e. pq) is the product of the constant terms in the factored form (i.e. the p and the q). 3. The linear coefficient in each trinomial (i.e. p + q) is the sum of the constant terms in the factored form. Steps to Factor the Quadratic Expression 1x 2 + bx + c = x 2 + bx + c = (x + p)(x + q) 1. List all the pairs of integers whose product is c. 2. Out of the list from step 1 find the pair of integers whose sum is b; those two integers are p and q. 3. Write the factored form of the expression: (x + p)(x + q).

17 6.3 Factoring Quadratic Polynomials (Page 17 of 33) Example 1 1. Factor x 2 + 8x The graph of f (x) = x 2 + 8x +12 is shown. What are the x-intercepts in the graph of f? 3. Write f in factored form. If f (r) = 0, then r is a zero of f. That is, a zero of a function is the input number that makes the output number zero. To find a zero of a function, set the function equal to zero and solve. Notice that if r is a zero, then (r, 0) is the x-intercept. 4. Find the zeros of f. Example 2 1. Factor x 2! 8x +12 y = x 2! 8x Factor x 2! x! 20 y = x 2! x! Factor x 2 + x! 20 y = x 2 + x! 20

18 6.3 Factoring Quadratic Polynomials (Page 18 of 33) Example 3 1. Factor a 2 +12a Factor p 2! 9 p Factor c 2! 3c! Factor y 2! 21y! 72 When the Quadratic Coefficient is not One If the coefficient on the quadratic term is not one, then the quadratic expression must be treated differently. The first thing to consider is whether or not there is a common factor in all of the terms of the expression that can be factored out. Example 4 Factor Out the GCF Factor [completely]. 1. 8x 3! 32x 2. 5x x x

19 6.3 Factoring Quadratic Polynomials (Page 19 of 33) Example 5 Factor 1. Factor!24x x 3! 27x 2 2. Factor!9y y! Factor 3x 2!15x Factor 5x 2!15x!140

20 6.4 Factoring Polynomials (Page 20 of 33) 6.4 Factoring Polynomials Example 4 Factor by Grouping (4-term polynomials) 1. Factor x 3! 2x 2 + 5x!10 2. Factor 10x 3! 6x 2 + 5x! 3 Steps to Factor the Quadratic Expression ax 2 + bx + c 1. List all pairs of integers whose product is ac. Out of that list find the pair of integers whose sum is b; call those two integers m and n so that b = m + n. 2. Rewrite the bx term as mx + nx so that ax 2 + bx + c = ax 2 + mx + nx + c 3. Group the first two terms and the last two terms of the fourterm expression and factor out the greatest common factor from each pair of terms. 4. Factor out the common binomial factor from the resulting expression. Example 5 Factor 3x 2! x! 4

21 6.4 Factoring Polynomials (Page 21 of 33) Example 6 1. Factor 10x 2! x! 3 2. Factor 3a 2!13a Factor 6z 2! 7z Factor 18y 2! 27 y Factor 15x 2! 22x + 8

22 6.5 Factoring Special Polynomials (Page 22 of 33) 6.5 Factoring Special Polynomials Example 1 1. Multiply (x + 5)(x! 5) 2. Multiply (a + b)(a! b) Factoring the Difference of Two Squares The Difference of Two Squares Example 2 1. Factor c 2! 64 The Product of a Sum and Difference of Two Terms A 2! B 2 = ( A + B)( A! B) 2. Factor 4x 2! Factor 18! 2d 2 4. Factor 16x 2! 36

23 6.5 Factoring Special Polynomials (Page 23 of 33) Factoring the Sum or Difference of Two Cubes A 3 + B 3 = ( A + B)( A 2! AB + B 2 ) Sum of two cubes A 3! B 3 = ( A! B)( A 2 + AB + B 2 ) Difference of two cubes Example 2 Factor 1. x x 3! t w x 5! 24x 2 y 3

24 6.5 Factoring Special Polynomials (Page 24 of 33) Example 3 Factor x 6! y 6 Factoring Guidelines / Strategies 1. If the GCF is not 1, then factor it out. 2. If a binomial is a difference of two square, then use a 2! b 2 = (a + b)(a! b) 3. For a four-term polynomial apply factor by grouping. 4. For a trinomial ax 2 + bx + c : a. If a = 1, then try to find two integers p and q whose product is c and whose sum is b. If the two numbers exist, then the factored form of the trinomial is (x + p)(x + q). b. If a! 1, then try to find two integers p and q whose product is ac and whose sum is b. If the two numbers exist, then rewrite bx = px + qx and factor the new fourterm polynomial by grouping. 5. Repeat all steps until no factors can be factored any further. Example 4 Factor Completely Factor [completely]. 1. 8! 2x 2 + x 3! 4x 2. x 2 +15x + 56

25 6.5 Factoring Special Polynomials (Page 25 of 33) Example 5 Factor Completely Factor [completely]. 1. 5x! 25x x 2! 22x 3 + 8x 4 3. x 4! 2x x! x 2!15x + 40x t 2 w 2! 8w a 3 b! 21a 2 b 2 +18ab 3

26 6.6 Solving polynomial equations by factoring (Page 26 of 33) 6.6 Solving Polynomial Equations by Factoring Quadratic Equation in One Variable A quadratic equation in one variable can be written in the form ax 2 + bx + c = 0, where a! 0, ax 2 is the quadratic term, bx is the linear term, and c is the constant term. Zero Factor Property ab = 0 if and only if a = 0 or b = 0 In words this states that a product is zero if and only if one of the factors is zero. To use this principle to solve equations, two requirements are necessary: (1) one side of the equation must be zero, and (2) the other side must be in factored form. Example 1 1. Solve (x + 5)(x! 3) = 0 2. Solve c(2c + 7) = 0 3. Solve x 2! 4x! 5 = 0 4. Solve 4x 2!11x + 7 = 0

27 6.6 Solving polynomial equations by factoring (Page 27 of 33) Example 2 1. Solve 4y! 7 =!3y 2 2. Solve (x + 2)(x! 4) = 7 3. Solve 25x 2 = Solve 1 4 x2 = 1 2 x + 2

28 6.6 Solving polynomial equations by factoring (Page 28 of 33) Zero of a Function If f (r) = 0, then r is a zero of f. That is, a zero of a function is the input number that makes the output number zero. To find a zero of a function, set the function equal to zero and solve. Example 3 1. Find the zeros of f (x) = x 2! 7x +10. f 2. Find the x-intercept(s) in the graph of f. 3. Write f in factored form. 4. Summarize your results: Zeros of f x-intercepts of f Factored form of f Equivalence of Zeros, x-intercepts, and Factors Let f (x) = ax 2 + bx + c, a! 0, and r be a real number. Then the following statements are equivalent. 1. f (r) = 0 read r is a zero of f 2. (r, 0) is an x-intercept in the graph of f. 3. (x! r) is a factor of f.

29 6.6 Solving polynomial equations by factoring (Page 29 of 33) Example 5 Let f (x) = x 2! 9x Write f in factored form. f 2. Find the following: Zeros of f x-intercepts of f Factored form of f 3. Find f (2). 4. Find the value(s) of x so that f (x) = The domain of f written in interval notation is 6. The range of f written in interval notation is

30 6.6 Solving polynomial equations by factoring (Page 30 of 33) Cubic Equation in One Variable A cubic equation in one variable can be written in the form ax 3 + bx 2 + cx + d = 0, where a! 0. The cubic term is ax 3, the quadratic term is bx 2, the linear term is cx and the constant term is d. Example 5 1. Solve 2x 3 = 42x + 8x Find the x-intercepts in the graph of f (x) = 2x 3! 8x 2! 42x y = 2x 3! 8x 2! 42x Example 6 1. Solve x 3! 5x 2! 4x + 20 = Find the x-intercepts in the graph of f (x) = x 3! 5x 2! 4x + 20 y = x 3! 5x 2! 4x + 20

31 6.6 Solving polynomial equations by factoring (Page 31 of 33) Facts About Cubic Equations and Functions 1. The cubic equation ax 3 + bx 2 + cx + d = 0 can have one, two or three real solutions. 2. The cubic function f (x) = ax 3 + bx 2 + cx + d can have one, two or three real zeros. 3. The graph of a cubic function f (x) = ax 3 + bx 2 + cx + d can have one, two or three x-intercepts. 4. The graph of a cubic function f (x) = ax 3 + bx 2 + cx + d can have either two turning points or no turning points. 5. The graph of a cubic function must either rise on the far left and fall on the far right, or fall on the far left and rise on the far right. 5. The domain of all cubic functions is all real numbers. 6. The range of all cubic functions is all real numbers. y =!0.5(x! 3) 3 y = 0.25(x + 6)(x! 3) 2 y = (x + 2)(x! 2)(x! 5)

32 6.6 Solving polynomial equations by factoring (Page 32 of 33) Example 7 Method 1 Solve on your graphing calculator x 2! x! 7 =!x Set Y 1 = x 2! x! 7 left side of the equation Y 2 =!x 2 right side of the equation 2. Since we want to know where the left side of the equation equals the right side, use the intersect program to find the points of intersection of the two functions. The x-values of the points of intersection are the solution to the original equation. 3. Check your solutions in the original equation. Example 7 Method 2 Solve on your graphing calculator x 2! x! 7 =!x Rewrite the equation so there is a zero on one side. x 2! x! 7 =!x 2 2x 2! x! 7 = 0 2. Set Y 1 = f (x) = 2x 2! x! 7 and find the zeros of f. 3. Check your solutions in the original equation.

33 6.6 Solving polynomial equations by factoring (Page 33 of 33) Example 10 The annual revenues for American Express are shown in the table. Let r(t) be the revenue (in billions of dollars) of American Express at t years since Find the appropriate regression equation (linear, exponential or quadratic) that models the data well. r(t) = 2. Predict the revenue in Year Revenue (billions of dollars) Predict when the revenue will be $39.7 billion. Example 11 A person has a rectangular garden with a width of 9 feet and a length of 12 feet. She plans to place mulch outside of the garden to form a border of uniform width. She has just enough mulch to cover 100 square feet of land. Determine the length and width of the garden with its mulch border.

1.3 Polynomials and Factoring

1.3 Polynomials and Factoring 1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.

More information

Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).

Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32

More information

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method. A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

More information

Factoring Polynomials

Factoring Polynomials UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

More information

NSM100 Introduction to Algebra Chapter 5 Notes Factoring

NSM100 Introduction to Algebra Chapter 5 Notes Factoring Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

More information

Section 6.1 Factoring Expressions

Section 6.1 Factoring Expressions Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what

More information

Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF

Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials

More information

Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

More information

( ) FACTORING. x In this polynomial the only variable in common to all is x.

( ) FACTORING. x In this polynomial the only variable in common to all is x. FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated

More information

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

More information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

1.3 Algebraic Expressions

1.3 Algebraic Expressions 1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

More information

Factoring Polynomials

Factoring Polynomials Factoring a Polynomial Expression Factoring a polynomial is expressing the polynomial as a product of two or more factors. Simply stated, it is somewhat the reverse process of multiplying. To factor polynomials,

More information

Higher Education Math Placement

Higher Education Math Placement Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

More information

6.3 FACTORING ax 2 bx c WITH a 1

6.3 FACTORING ax 2 bx c WITH a 1 290 (6 14) Chapter 6 Factoring e) What is the approximate maximum revenue? f) Use the accompanying graph to estimate the price at which the revenue is zero. y Revenue (thousands of dollars) 300 200 100

More information

Factoring Polynomials

Factoring Polynomials Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.

Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method

More information

Factoring Quadratic Expressions

Factoring Quadratic Expressions Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the

More information

1.1 Practice Worksheet

1.1 Practice Worksheet Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)

More information

7.1 Graphs of Quadratic Functions in Vertex Form

7.1 Graphs of Quadratic Functions in Vertex Form 7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called

More information

Algebra and Geometry Review (61 topics, no due date)

Algebra and Geometry Review (61 topics, no due date) Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

More information

FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

More information

Name Intro to Algebra 2. Unit 1: Polynomials and Factoring

Name Intro to Algebra 2. Unit 1: Polynomials and Factoring Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332

More information

POLYNOMIALS and FACTORING

POLYNOMIALS and FACTORING POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use

More information

Factoring Trinomials: The ac Method

Factoring Trinomials: The ac Method 6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For

More information

Factoring a Difference of Two Squares. Factoring a Difference of Two Squares

Factoring a Difference of Two Squares. Factoring a Difference of Two Squares 284 (6 8) Chapter 6 Factoring 87. Tomato soup. The amount of metal S (in square inches) that it takes to make a can for tomato soup is a function of the radius r and height h: S 2 r 2 2 rh a) Rewrite this

More information

Veterans Upward Bound Algebra I Concepts - Honors

Veterans Upward Bound Algebra I Concepts - Honors Veterans Upward Bound Algebra I Concepts - Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER

More information

Factoring and Applications

Factoring and Applications Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the

More information

Mathematics Placement

Mathematics Placement Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

More information

Chapter R.4 Factoring Polynomials

Chapter R.4 Factoring Polynomials Chapter R.4 Factoring Polynomials Introduction to Factoring To factor an expression means to write the expression as a product of two or more factors. Sample Problem: Factor each expression. a. 15 b. x

More information

MATH 90 CHAPTER 6 Name:.

MATH 90 CHAPTER 6 Name:. MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a

More information

HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

More information

Algebra II A Final Exam

Algebra II A Final Exam Algebra II A Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. Evaluate the expression for the given value of the variable(s). 1. ; x = 4 a. 34 b.

More information

Factoring Polynomials and Solving Quadratic Equations

Factoring Polynomials and Solving Quadratic Equations Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3

More information

Operations with Algebraic Expressions: Multiplication of Polynomials

Operations with Algebraic Expressions: Multiplication of Polynomials Operations with Algebraic Expressions: Multiplication of Polynomials The product of a monomial x monomial To multiply a monomial times a monomial, multiply the coefficients and add the on powers with the

More information

CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

More information

Polynomials and Quadratics

Polynomials and Quadratics Polynomials and Quadratics Want to be an environmental scientist? Better be ready to get your hands dirty!.1 Controlling the Population Adding and Subtracting Polynomials............703.2 They re Multiplying

More information

Mathematics Curriculum

Mathematics Curriculum Common Core Mathematics Curriculum Table of Contents 1 Polynomial and Quadratic Expressions, Equations, and Functions MODULE 4 Module Overview... 3 Topic A: Quadratic Expressions, Equations, Functions,

More information

Algebra 2 PreAP. Name Period

Algebra 2 PreAP. Name Period Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing

More information

1.3. Maximum or Minimum of a Quadratic Function. Investigate A

1.3. Maximum or Minimum of a Quadratic Function. Investigate A < P1-6 photo of a large arched bridge, similar to the one on page 292 or p 360-361of the fish book> Maximum or Minimum of a Quadratic Function 1.3 Some bridge arches are defined by quadratic functions.

More information

Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only

Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials

More information

Tool 1. Greatest Common Factor (GCF)

Tool 1. Greatest Common Factor (GCF) Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When

More information

SPECIAL PRODUCTS AND FACTORS

SPECIAL PRODUCTS AND FACTORS CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

More information

FACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1

FACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1 5.7 Factoring ax 2 bx c (5-49) 305 5.7 FACTORING ax 2 bx c In this section In Section 5.5 you learned to factor certain special polynomials. In this section you will learn to factor general quadratic polynomials.

More information

Factor Polynomials Completely

Factor Polynomials Completely 9.8 Factor Polynomials Completely Before You factored polynomials. Now You will factor polynomials completely. Why? So you can model the height of a projectile, as in Ex. 71. Key Vocabulary factor by grouping

More information

By reversing the rules for multiplication of binomials from Section 4.6, we get rules for factoring polynomials in certain forms.

By reversing the rules for multiplication of binomials from Section 4.6, we get rules for factoring polynomials in certain forms. SECTION 5.4 Special Factoring Techniques 317 5.4 Special Factoring Techniques OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor

More information

FACTORING POLYNOMIALS

FACTORING POLYNOMIALS 296 (5-40) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated

More information

Polynomial Expressions and Equations

Polynomial Expressions and Equations Polynomial Expressions and Equations This is a really close-up picture of rain. Really. The picture represents falling water broken down into molecules, each with two hydrogen atoms connected to one oxygen

More information

Using the ac Method to Factor

Using the ac Method to Factor 4.6 Using the ac Method to Factor 4.6 OBJECTIVES 1. Use the ac test to determine factorability 2. Use the results of the ac test 3. Completely factor a trinomial In Sections 4.2 and 4.3 we used the trial-and-error

More information

Introduction Assignment

Introduction Assignment PRE-CALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying

More information

Algebra 1 Course Title

Algebra 1 Course Title Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

More information

MATH 10034 Fundamental Mathematics IV

MATH 10034 Fundamental Mathematics IV MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

More information

Section 3.1 Quadratic Functions and Models

Section 3.1 Quadratic Functions and Models Section 3.1 Quadratic Functions and Models DEFINITION: A quadratic function is a function f of the form fx) = ax 2 +bx+c where a,b, and c are real numbers and a 0. Graphing Quadratic Functions Using the

More information

PARABOLAS AND THEIR FEATURES

PARABOLAS AND THEIR FEATURES STANDARD FORM PARABOLAS AND THEIR FEATURES If a! 0, the equation y = ax 2 + bx + c is the standard form of a quadratic function and its graph is a parabola. If a > 0, the parabola opens upward and the

More information

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

More information

Sect 6.7 - Solving Equations Using the Zero Product Rule

Sect 6.7 - Solving Equations Using the Zero Product Rule Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred

More information

Factoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai

Factoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai Factoring Guidelines Greatest Common Factor Two Terms Three Terms Four Terms 008 Shirley Radai Greatest Common Factor 008 Shirley Radai Factoring by Finding the Greatest Common Factor Always check for

More information

Determinants can be used to solve a linear system of equations using Cramer s Rule.

Determinants can be used to solve a linear system of equations using Cramer s Rule. 2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution

More information

Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

More information

FACTORING OUT COMMON FACTORS

FACTORING OUT COMMON FACTORS 278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the

More information

Polynomial Operations and Factoring

Polynomial Operations and Factoring Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.

More information

LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

More information

A Systematic Approach to Factoring

A Systematic Approach to Factoring A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool

More information

In algebra, factor by rewriting a polynomial as a product of lower-degree polynomials

In algebra, factor by rewriting a polynomial as a product of lower-degree polynomials Algebra 2 Notes SOL AII.1 Factoring Polynomials Mrs. Grieser Name: Date: Block: Factoring Review Factor: rewrite a number or expression as a product of primes; e.g. 6 = 2 3 In algebra, factor by rewriting

More information

Understanding Basic Calculus

Understanding Basic Calculus Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

More information

Factoring, Solving. Equations, and Problem Solving REVISED PAGES

Factoring, Solving. Equations, and Problem Solving REVISED PAGES 05-W4801-AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring

More information

ALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals

ALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an

More information

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial

More information

Some Lecture Notes and In-Class Examples for Pre-Calculus:

Some Lecture Notes and In-Class Examples for Pre-Calculus: Some Lecture Notes and In-Class Examples for Pre-Calculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax

More information

CRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide

CRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are

More information

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

More information

Greatest Common Factor (GCF) Factoring

Greatest Common Factor (GCF) Factoring Section 4 4: Greatest Common Factor (GCF) Factoring The last chapter introduced the distributive process. The distributive process takes a product of a monomial and a polynomial and changes the multiplication

More information

Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions:

Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions: Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?

More information

AIP Factoring Practice/Help

AIP Factoring Practice/Help The following pages include many problems to practice factoring skills. There are also several activities with examples to help you with factoring if you feel like you are not proficient with it. There

More information

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

More information

What are the place values to the left of the decimal point and their associated powers of ten?

What are the place values to the left of the decimal point and their associated powers of ten? The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

More information

Academic Success Centre

Academic Success Centre 250) 960-6367 Factoring Polynomials Sometimes when we try to solve or simplify an equation or expression involving polynomials the way that it looks can hinder our progress in finding a solution. Factorization

More information

MATH 21. College Algebra 1 Lecture Notes

MATH 21. College Algebra 1 Lecture Notes MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a

More information

Factor and Solve Polynomial Equations. In Chapter 4, you learned how to factor the following types of quadratic expressions.

Factor and Solve Polynomial Equations. In Chapter 4, you learned how to factor the following types of quadratic expressions. 5.4 Factor and Solve Polynomial Equations Before You factored and solved quadratic equations. Now You will factor and solve other polynomial equations. Why? So you can find dimensions of archaeological

More information

5.1 FACTORING OUT COMMON FACTORS

5.1 FACTORING OUT COMMON FACTORS C H A P T E R 5 Factoring he sport of skydiving was born in the 1930s soon after the military began using parachutes as a means of deploying troops. T Today, skydiving is a popular sport around the world.

More information

Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c

Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics

More information

Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper

Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic

More information

Algebra Cheat Sheets

Algebra Cheat Sheets Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

More information

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions. Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

More information

Algebra 1 Chapter 08 review

Algebra 1 Chapter 08 review Name: Class: Date: ID: A Algebra 1 Chapter 08 review Multiple Choice Identify the choice that best completes the statement or answers the question. Simplify the difference. 1. (4w 2 4w 8) (2w 2 + 3w 6)

More information

This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide.

This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the

More information

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with

More information

MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

More information

DRAFT. Algebra 1 EOC Item Specifications

DRAFT. Algebra 1 EOC Item Specifications DRAFT Algebra 1 EOC Item Specifications The draft Florida Standards Assessment (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

More information

Warm-Up Oct. 22. Daily Agenda:

Warm-Up Oct. 22. Daily Agenda: Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment

More information

This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0).

This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/

More information

Lesson 9.1 Solving Quadratic Equations

Lesson 9.1 Solving Quadratic Equations Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte

More information

Answer Key for California State Standards: Algebra I

Answer Key for California State Standards: Algebra I Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

More information

0.4 FACTORING POLYNOMIALS

0.4 FACTORING POLYNOMIALS 36_.qxd /3/5 :9 AM Page -9 SECTION. Factoring Polynomials -9. FACTORING POLYNOMIALS Use special products and factorization techniques to factor polynomials. Find the domains of radical expressions. Use

More information

Solving Quadratic Equations

Solving Quadratic Equations 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

More information

Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

More information