# 6.1 Add & Subtract Polynomial Expression & Functions

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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.

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