# Topic 7: Polynomials. Introduction to Polynomials. Table of Contents. Vocab. Degree of a Polynomial. Vocab. A. 11x 7 + 3x 3

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1 Topic 7: Polynomials Table of Contents 1. Introduction to Polynomials. Adding & Subtracting Polynomials 3. Multiplying Polynomials 4. Special Products of Binomials 5. Factoring Polynomials 6. Factoring Polynomials, part 7. Factoring by Grouping Introduction to Polynomials Vocab Monomial: a number, a variable, or a product of numbers and variables with whole number exponents. Degree of a monomial: is the sum of the exponents of the variables. A constant has degree 0. Vocab Polynomial: an expression of more than two algebraic terms. Example: 3x 4 + 5x 7x + 1 Degree of a polynomial is the degree of the term with the greatest power/exponent. Find the degree of each polynomial. A. 11x 7 + 3x 3 B. Degree of a Polynomial Example: The degree of 3x 4 + 5x 7x + 1 is 4. 1

2 Vocab Standard form of a polynomial: Polynomial written with the terms in order from greatest degree to least degree. Leading Coefficient: When written in standard form, the coefficient of the first term is called the leading coefficient. Example: 3x 4 + 5x 7x + 1 and 3 is the leading coefficient. Write the polynomial in standard form. Then give the leading coefficient. 1. 6x 7x 5 + 4x x + x 5 + 9x y 5 3y y Special Polynomial Names By Degree Degree Name 0 Constant 1 Linear Quadratic 3 Cubic 4 Quartic 5 Quintic By # of Terms Terms Name 1 Monomial Binomial 3 Trinomial 4 or more Polynomial Review: Like Terms You can add or subtract monomials by adding or subtracting like terms. Like terms 4a 3 b + 3a b 3 a 3 b Not like terms The variables have the same powers. The variables have different powers. 6 or more 6 th,7 th,degree and so on Identify Like Terms Identify the like terms in each polynomial. A. 5x 3 + y + 6y + 4x 3 Like terms: B. 3a 3 b + 3a b 3 + a 3 b a 3 b Like terms:

3 Adding and Subtracting Polynomials Adding and Subtracting Polynomials Just as you can perform operations on numbers, you can perform operations on polynomials. To add or subtract polynomials, combine like terms. Remember! Like terms are constants or terms with the same variable(s) raised to the same power(s). Simplifying Polynomials Combine like terms. A. 1p p + 8p 3 Combine like terms. c. s + 3s + s 3s 5s B. 5x 6 3x + 8 d. 4z z +16z z 3 7 Methods: Adding Polynomials Polynomials can be added in either vertical or horizontal form. In vertical form, align the like terms and add: 5x + 4x +1 + x + 5x + 7x +9x +3 In horizontal form, use the Associative and Commutative Properties to regroup and combine like terms. (5x + 4x + 1) + (x + 5x + ) = (5x + x + 1) + (4x + 5x) + (1 + ) = 7x + 9x + 3 Adding Polynomials Add. A. (4m + 5) + (m m + 6) B. (10xy + x) + ( 3xy + y) 3

4 Add. Add (5a 3 + 3a 6a + 1a ) + (7a 3 10a). Subtracting Polynomials To subtract polynomials, remember that subtracting is the same as adding the opposite (distributing the negative). To find the opposite of a polynomial, you must write the opposite of each term in the polynomial: Subtracting Polynomials Subtract. ( 10x 3x + 7) (x 9) (x 3 3x + 7)= x 3 + 3x 7 Subtracting Polynomials Subtract. (x 3 + 4y) (x 3 ) Subtract. (9q 3q) (q 5) (7m 4 m ) (5m 4 5m + 8) (x 3x + 1) (x + x + 1) 4

5 5

6 Multiplying Polynomials F.O.I.L Multiplying Polynomials Each term in the first polynomial, must be multiplied by each term in the second polynomial. Method 1: Distribute F.O.I.L. First Outer Inner Last Multiply!!! Multiply (3x + 4)(5x 3) 1. Draw a box.. Write a polynomial on the top and side of a box. 3. Multiply. 4. Combine like terms. Method : Box 5x -3 3x +4 Pick Your Method: (7p )(3p 4) First terms: 1p Outer terms: -8p Inner terms: -6p Last terms: +8 Combine like terms. 1p 34p + 8 3p -4 7p - 1p -8p -6p +8 6

7 1. (7x 10)(3x + 8). (x 3)(4x 8) 3. (5x 10)(x +8) Multiplying Terms with Exponents When FOILing, add the exponents and multiply coefficients. Add the little numbers and multiply the big numbers!!! Example: (3x + 10x)(5x 3 7x ) 15x 5 1x x 4 70x 3 15x 5 + 9x 4 70x 3 1. (7x 10x)(3x 3 + 8x ). (x 4 3x )(4x 8) Multiplying Larger Polynomials Each term in the 1 st polynomial must be multiplied by each term in the nd. Example: (7x + x + 8)(4x 3 9x ) 3. (5x 3 + x )(8x 7) Method : Multiply: (x 5)(x 5x + 4) 1. (5x + 7) (x 3 5x +9) 7

8 . (10x 4 5x + 8) (8x 3 3x 6) 8

9 Special Products of Binomials 1. (x + 3)(x + 3). (3x 4)(3x + 4) 3. (x + 5)(x + 5) Multiply: 4. (6x 1)(6x 1) 5. (3x + y) Multiply: Special Products of Binomials Name Algebraically Words Positive Perfect Square (a + b) Negative Perfect Square (a b) Difference of Squares (a + b)(a b). 1. (x + 8). (4x + 6y) 3. ( x + 5) 4. (7x 3)(7x + 3) 9

10 1. (x 3) Let s Practice #. (x + 4y) 3. (x + 5)(x 5) 4. (x + 4) 10

11 Factoring Polynomials Vocab: Factoring Factoring is rewriting an expression as a product of factors. It is the reverse of multiplying polynomials FOILing. x bx c To determine the factors, ask yourself What two # s add to the middle number AND multiply to the last number?!?! x What adds (or subtracts) to get 3 and multiplies to 3x get? Factor: 1. x + 5x + 6 What adds (or subtracts) to get 7 and x 7x 10 multiplies to get 10?. x 7x + 10 x 7x 44 What adds (or subtracts) to get 7 and multiplies to get 44? 3. x 11x +4 11

12 Signs of Factors x bx c b c Factors + + +,+ + +, +, (The factor w/ the greater absolute value is ) +, (The factor w/ the greater absolute value is +) Vocab: GCF The greatest common factor (GCF) is a common factor of the terms in the expression. Example: 9x 9x 18 4x 8x 1 Vocab: Prime If a polynomials is prime it means there are no factors. Find the prime polynomials below: 1. x + 7x + 9. x + 5x x + 9x + 10 Factor. 1. y 10y +16. r 11r n 15n v + 10v x + 1x x 8x x +x x 1

13 Factoring Polynomials, Part Expanded Form Expanded Form (x 1)(3x 5) 6x 10x 3x 5 6x 13x 5 When factoring problems where a 1, we first want to get the problem into expanded form before we try to factor. Creating Expanded Form Step 1: Multiply a c Step : To get to expanded form ask yourself What multiplies to get a c, and add/subtracts to get to b. Example: 1. Expand: x +9x +7. Expand: 3x + x 8 Method 1: Step 3: Write your new factors in place of bx. Step 4: Group the first two terms together and the last two terms together. Step 5: Factor each group Step 6: Factor again to get the complete factorization Method 1: 6x + 13 x +5 1) Multiply a c (6 5=30) ) To get to expanded form ask yourself What multiplies to get a c, and add/subtracts to get to b. (10, 3) 3) Write your new factors in place of bx. (6x +10x+3x+5) 4) Group the first two terms together and the last two terms together. [(6x +10x)+(3x+5)] 5) Factor each group [x(3x+5)+1(3x+5)] 6) Factor again to get the complete factorization [(3x+5)(x+1)] Method : 6x + 13 x +5 Step 3: Fill in box. Step 4: Factor horizontally and vertically. Step 5: Terms outside of box are the solution. Original 1 st Term Expanded Term 1 Expanded Term Original Last Term 13

14 Factor: x + 5x 1 Factor: 3x + 7x + Original 1 st Term Expanded Term 1 Expanded Term Original Last Term Factor: x + 15x 8 Factor: 16x + 8x +10 Factoring by Grouping Factoring by Grouping Using the distributive property to factor polynomials with four or more terms. Terms can be put into groups and then factored each group will have a like factor used in regrouping. 14

15 Factoring by Grouping A polynomial can be factored by grouping if all of the following conditions exist. 1. There are four or more terms.. Terms have common factors that can be grouped together, and 3. There are two common factors that are identical. Factor by Grouping Factor each polynomial by grouping. Check your answer. 6h 4 4h 3 + 1h 8 Symbols: ax + bx + ay + by = (ax + bx) + (ay + by) Group, factor Regroup = x(a + b) + y(a + b) = (x + y)(a + b) Factor by Grouping Factor each polynomial by grouping. 5y 4 15y 3 + y 3y Factor each polynomial by grouping. 6b 3 + 8b + 9b + 1 Factor each polynomial by grouping. 4r 3 + 4r + r + 6 Factoring with Opposite Groups x 3 1x x 15

16 Factor each polynomial. Check your answer. 15x 10x 3 + 8x 1 Factor each polynomial by grouping. 1. x 3 + x 6x 3. 7p 4 p p 18 Factoring Procedure 16

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