A-2. Polynomials and Factoring. Section A-2 1

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1 A- Polynomials and Factoring Section A- 1

2 What you ll learn about Adding, Subtracting, and Multiplying Polynomials Special Products Factoring Polynomials Using Special Products Factoring Trinomials Factoring by Grouping... and why You need to review these basic algebraic skills if you don t remember them. Section A-

3 ADDING, SUBTRACTING, AND MULTIPLYING POLYNOMIALS A polynomial in x is any expression that can be written in the form a n n 1 n x + an 1 x + a1x + a0, where n is a nonnegative integer, and a n 0. Coefficients are the numbers Degree of the polynomial is n Leading Coefficient is a n a n 1,..., a a 1, 0 Section A-

4 ADDING, SUBTRACTING, AND MULTIPLYING POLYNOMIALS A polynomial with one term is called a monomial. A polynomial with two terms is called a binomial. A polynomial with three terms is called a trinomial. Standard form is a polynomial written with powers of x in descending order. Like terms have the same variable each raised to the same power. Section A- 4

5 Example 1: Adding and Subtracting Polynomials a) (x x + 4x 1) + ( x + x 5x + ) b) (4x + x 4) (x + x x + ) Section A- 5

6 THE DISTRIBUTIVE PROPERTY To expand the product of two polynomials we use the distributive property. (Formerly known as FOILing) ( x + )(4x 5) Section A- 6

7 THE DISTRIBUTIVE PROPERTY Multiplying two polynomials requires multiplying each term of one polynomial by every term of the other polynomial. Two ways to do this: Distribute every term in the 1 st polynomial to every term in the nd polynomial Multiply in vertical form Let s take a look at both, you choose your favorite way. Section A- 7

8 Example a: MULTIPLYING POLYNOMIALS ( )( ) Write x 4x + x + 4x + 5 in standard form. Section A- 8

9 Example b: MULTIPLYING POLYNOMIALS ( )( ) Write x 4x + x + 4x + 5 in standard form. Section A- 9

10 SPECIAL PRODUCTS Let u and v be real numbers, variables, or algebraic expressions. 1. Product of a sum and a difference: 1. Square of a sum:. Square of a difference:. Cube of a sum: 4. Cube of a difference: ( u + v)( u v) = u v ( u + v) = u + uv + v ( u v) = u uv + v ( u + v) = u + u v + uv + v ( u v) = u u v + uv v Section A- 10

11 a) Example : USING SPECIAL PRODUCTS ( x + 8)( x 8) b) ( 5y 4) c) ( ) x y Section A- 11

12 FACTORING POLYNOMIALS USING SPECIAL PRODUCTS When we write a polynomial as a product of two or more polynomial factors we are factoring a polynomial. Unless specified otherwise, we factor polynomials into factors of lesser degree and with integer coefficients. A polynomial that cannot be factored using integer coefficients is a prime polynomial. Section A- 1

13 FACTORING POLYNOMIALS USING SPECIAL PRODUCTS A polynomial is completely factored if it is written as a product of its prime factors. For example, x + x + x + 1 = ( x + 1)( x Is completely factored + 1) However, x 9x = x( x 9) Is not because of the difference of squares. u squares v difference = ( u + v)( u v) Section A- 1

14 Example 4: REMOVING COMMON FACTORS The FIRST step in factoring (ALWAYS) is to take out common factors (using the distributive property (backwards)). a) x + x + 6 x ( b + c) = ab ac a + b) u v + uv Section A- 14

15 Example 5: FACTORING THE DIFFERENCE OF TWO SQUARES a) x 64 b) 49x 1 Section A- 15

16 Example 6: FACTORING PERFECT SQUARE TRINOMIALS 9x + 6x + 1 ( u + v) = u + uv + v Section A- 16

17 Example 7: FACTORING THE SUM AND DIFFERENCE OF TWO CUBES In the sum and difference of two cubes, notice the pattern of the signs. same sign same sign ( )( ) u + v u uv v + v = ( )( ) u v = u v u + uv + v u + opposite sign opposite sign a) x 7 b) 64 x + 8 Section A- 17

18 FACTORING TRINOMIALS Factoring the trinomial ax + bx + c into a product of binomials with integer coefficients requires factoring the integers a and c. Factors of a ax ( )( ) bx + c = x x Factors of c Section A- 18

19 Example 8: FACTORING A TRINOMIAL WITH LEADING COEFFICIENT = 1 r + 9r + 14 Section A- 19

20 Example 9: FACTORING A TRINOMIAL WITH LEADING COEFFICIENT 1 5y y 1 Section A- 0

21 Example 10: FACTORING TRINOMIALS IN X AND Y x 7xy + y Section A- 1

22 Example 11: FACTORING BY GROUPING a) x + x 6x b) ac ad + bc bd Section A-

23 Factoring Polynomials CHECKLIST 1. Look for common factors.. Look for special polynomial forms.. Factor using guess and check, grouping, box, No Fuss, etc. 4. If there are four terms, try grouping. Section A-

24 HOMEWORK Section A- (page 851) 1- odd, odd, 59-7 all, all Section A- 4