Mth 95 Module 2 Spring 2014


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1 Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression are separated by addition and subtraction signs. Examples: A monomial is a term, which has only nonnegative integer exponents. (No addition or subtraction signs no division by variables) Monomials Not Monomials The degree of a monomial is the sum of the exponents of its variables. A constant has degree 0, unless the term is 0, which has undefined degree. Give the degree of each monomial. 3 x 17 3 x yz 0 A coefficient is the numerical constant (including its sign) in a monomial. Give the coefficient in each monomial 3 4 4x cd 8 A polynomial is a monomial or a sum of monomials. Polynomials Not Polynomials A binomial is a polynomial which has terms Examples: A trinomial is a polynomial that has terms. Examples: The degree of a polynomial equals the degree of the monomial with the highest degree. 4xy x 3 y 4 xy 6x 3 y + y 5 Degree of each term Polynomial s degree Chapter 5 1
2 Mth 95 Module Spring 014 The leading coefficient of a polynomial of one variable is the coefficient of the monomial with the highest degree Degree Type Example Leading Coefficient 0 Constant 14 1 Linear 3x 1 Quadratic 3y + 5y 4 3 Cubic x 3 4x x + 1 Standard form of a single variable polynomial is written in descending order of the powers of the variable. 3 3x6 7x 4n 5n 6 9n Standard form Degree Leading Term Leading Coefficient Number of Terms To add two polynomials combine like terms. Like terms contain the same variables raised to the same power. (So, x and x are NOT like terms.) Horizontal method Vertical method (3x 4x + 3) + (5x + 4x + 1) Try (3n + n 3 + 4) + (3n 3 8) To subtract two polynomials, add the first polynomial and the opposite of the second polynomial. The opposite of a polynomial is found by changing the sign of every term. (Remember, subtraction is NOT commutative.) (3x + 9) (x + 4) (x + 6) (x + 4x 1) Polynomial functions create smooth, continuous graphs. Remember: Since these are functions, they pass the vertical line test, i.e. for each input there is a unique output. Scale on the axes below is one. The horizontal axis is x; the vertical axis is f(x). x Type: f(x) f(x) = 3 f(x) = x 1 f(x) = x Chapter 5
3 Mth 95 Module Spring 014 f(x) = x 3 other Type: Symbolical Evaluating Polynomial Functions Evaluate f(x) = 4 x Evaluate f(x) = 3x 3 + x 4 at x = 1 and x = 3 at x = and x = 3 Which of the following represent polynomial functions? f(x) = x + 1 f(x) = x 3 f(x) = ½ + 3x x 5 f(x) = 4 x Multiplication of Polynomials Remember distribution 3(x 4) = (x 5y) = Properties of Exponents: For any nonzero numbers a and b and integers m and n, a m a n = a m + n (a m ) n = a mn (ab) n = a n b n Simplifying Polynomials 6x 4 4x 3 = (x y 3 )(3xy ) = 4m (m m 3 ) = xy(x 3 y) = (n 3 m) 3 = (xy ) 4 = 4y(y + 3y 5) = y 3 (y y + 6) = Multiplying Binomials Geometric area model Distribution (x + )(x + 1) (x + )(x + 1) = x (x + 1) + (x + 1) = x + x + x + = Chapter 5 3
4 Mth 95 Module Spring 014 Table Vertical FOIL a shortcut to multiply every term in x + the first binomial by every term in the x + 1 second binomial. (x + )(x + 1) Multiply the FIRST terms x x = x Multiply the OUTER terms x 1 = x Multiply the INNER terms x = x Multiply the LAST terms 1 = Write as a sum x + 3x + (3x )(x + 4) (5 x)(3 5x) (y 1)(y 3) x 7x 11 (n 5) (x 5)(x + 5) Multiplying other polynomials (x + 5)(x 4x + 5) Horizontal method Vertical method (3x )(x + 5x 4) x(x 4x + 5) + 5(x 4x + 5) x 4x + 5 x + 5 Special Products Multiplying binomials that are the sum and difference of the same two terms. ab ab a ab ab b ( difference of squares) y11 y 11 x3x 3 x4x 4 3x3x 1 1 x x (w + 5v )(w 5v ) Chapter 5 4
5 Mth 95 Module Spring 014 Squaring a binomial: Squaring a binomial sum: a b a ba b x 4 (3x + ) What if it was squaring a difference instead of a sum? a b a ba b x 3 x 3 Review: Simplify. Begin by multiplying to change each product to a sum x 3x x 3y 5x y 5y x x x Section 5.5 The Greatest Common Factor and Factoring by Grouping Factoring polynomials is important because we will use factoring to solve polynomial equations. Factoring is undistributing or unfoiling. It is the process of writing a polynomial as the product of polynomials. Distribution x (3x + 5) = 6x + 10x Foiling (x + 3) (x + ) = x + 5x + 6 Factoring 6x + 10x = x (3x + 5) Factoring x + 5x + 6 = (x + 3) (x + ) Factoring out a monomial greatest common factor Always look for the GCF before using other factoring patterns. 10x 3 6 x What factors are common to both terms? 5xxx 3xx Factor each monomial and find the GCF, the highest degree ( x x)(5x) (xx)(3) of each variable in common and the largest common coefficient. x (5x 3) Write as a product of the greatest common monomial and a polynomial Check your factoring by distribution Find the greatest common factor (GCF) y 7, 4y , 4, 8x 4 y 11x 3 y z 4, 33x y 5 z 6 Chapter 5 5
6 Mth 95 Module Spring 014 Factor the following to change each polynomial from a sum to a product. 6x 3 y + 9xy 1x 4 8x 8m 3 m + 6m (When the lead coefficient is negative, factor out a negative GCF.) 40x y + 15 x y x y 3 7xy x 3 6x + 4x Factoring out a binomial greatest common factor This looks a little messier: (x + 3) + x(x + 3) What is the common factor? ( + x) (x + 3) Write as a product. Factoring out a binomial greatest common factor 3 x 7 5y x 7 6x x 3y 5 x 3y x x 3x 3 x x x x 43x x 4 3x x x 4w w 3 w 3 6x x 3y x 3y Factoring by Grouping (Use this if you have four terms after you ve checked for a GCF of all four terms) 1. Group terms in pairs that have a common factor. Factor out the common monomial factor from each pair 3. Factor out, if it exists, the remaining common binomial factor 4. Check factors by multiplying xy 5x 9y x + 6x + 5x + 10 x x 5x Chapter 5 6
7 Mth 95 Module Spring 014 ab a 5b 10 15xy 0x 6y x xy x y xy 4y 3x x x xy y 3 x x x 3 6 Review: Multiply to change each product to a sum. Simplify x x 5 x 3y x 3y x x x 3y 4 Factor each polynomial using GCF or factoring by grouping x y +10x y 6xy 9x 8y 1 Section 5.6 Factoring Trinomials Remember factoring is undoing distribution. It is writing the sum as a product. Leading Coefficient is One: To factor x + bx + c, find integers m and n that satisfy m n = c and m + n = b, then x + bx + c = ( x + m)(x + n). For x + 6x + 8 ask, What pair of integer factors for 8 add to 6? Possible factor pairs are: 1 and 8, and 4, 1 and 8, and and 4 Sums: = = = = 6 Choose and 4 for m and n. Therefore, x + 6x + 8 = (x + )(x + 4). Check by foiling. If a polynomial cannot be factored, we say it is a prime polynomial. If c is positive, m and n are either both positive or both negative. If c is negative, m and n have opposite signs. x 8x 15 x 10x 4 a 8a 16 x 3x 8 y + 8y + 18 x xy 8y Chapter 5 7
8 Mth 95 Module Spring 014 Leading Coefficient is NOT One Sometimes a common factor must be factored out first before factoring the trinomial. Factor completely. 7x x + 4x m 4 + 6m 3 + 5m 3m 3 1m + 36m You can factor ax + bx + c, where a, b, and c have no common factor by the ac method or guess and check. The ac method: 1. Find numbers m and n such that mn = ac and m + n = b.. Write the trinomial as ax + mx + nx + c. 3. Use grouping to factor this expression as two binomials. Example x 5x 3 Since (3) = 6, ask what factors of 6 have a sum of 5. Since 6(1) = 6 and = 5 rewrite 5x as 6x + x x 6x + x 3 Next factor by grouping x(x 3) + 1 (x 3) (x + 1)(x 3) 10y + 13y 3 4n + 19n x 8x x 7x 6 y 5y 3 3x 13x 4 15x 39x 18 1x 5x 14x xy 3y Chapter 5 8
9 Mth 95 Module Spring 014 Section Factoring by Special Products Difference of Two Squares: For any real numbers a and b, a b = (a + b)(a b) Steps in identifying and factoring a difference of squares 1) Is it a binomial (two terms) difference (subtraction)? ) Are both terms perfect squares? (eg. 1, 4, 9, 16,,x, x 4,, 5x, ) 3) Take the square root of each term and substitute into the above pattern. 4) Check by foiling x 9 5x 16 4x + 9 9x 100 (x ) (n 4) y 144 9x 4y Binomials can only be factored by GCF and Difference of Squares. Always check for a GCF before using other factoring patterns. 6x 6y 3x 3 1xy 36x 4y 64z 4 49z Sometimes more than one factoring pattern must be applied or a pattern must be applied more than once. x 4 y 4 x 3 + x 9x 9 x 81y x 8x x Chapter 5 9
10 Mth 95 Module Spring 014 Perfect Square Trinomials: For and real numbers a and b, a + ab + b = (a + b) and a  ab + b = (a  b) Steps in identifying and factoring a difference of squares 1) Is it a trinomial (three terms) with the powers of the variable in order? ) Are the first and third terms POSITIVE perfect squares? 3) Multiply the square roots of these two terms and double the result. If it is a perfect square trinomial, the answer should be the same as or the opposite of the middle term of the polynomial. 4) Factor using one of the patterns above. 5) Check by foiling. x + x + 1 x 6x + 9 9y 6y 4 5x + 30xy + 9y 4z 4z 1 x x 1 x 0x x 16x 1 A General Factoring Strategy 1. For two or more terms, factor out any common factor..if you have only two terms, check for difference of squares. 3.If you have three terms, factor by trial and error, use the ac method or use one of the perfect square trinomial patterns. 4.If you have four terms, factor by grouping. 5.Continue factoring until none of the polynomial factors can be factored further. Factoring Flow Chart GCF Two terms: Use Difference of Squares a b = (a + b)(a b) Three terms: Use x + bx +c ax + bx + c, where a 1 a + ab + b = (a + b) a  ab + b = (a  b) Four terms: Use Factoring by grouping Chapter 5 10
11 Mth 95 Module Spring 014 Factor completely. 3x y 75y 3 x y 16y + 3 x 3y + 4y 6 4z 17z 4 4x 3x 60 18x + 30x x +5x 1 4x x 10 75a 48 x 5 3 n n n r 1r t 18rt 3 5x x 7 3xy 6y 5x 10 4r t 4 6 Chapter 5 11
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