1. Polynomials and Polynomial Functions
*Chapter 5
Polynomials and Polynomial Functions

2. Chapter Sections 5.1 – Addition and Subtraction of Polynomials
5.2 – Multiplication of Polynomials
5.3 – Division of Polynomials and Synthetic Division
5.4 – Factoring a Monomial from a Polynomial and Factoring by Grouping
5.5 – Factoring Trinomials
5.6 – Special Factoring Formulas
5.7-A General Review of Factoring
5.8- Polynomial Equations
Chapter 1 Outline

3. 5.7 A General Review of Factoring
a2 – b2 = (a + b)(a – b) a3 + b3 = (a + b)(a2 – ab + b2)
a3 – b3 = (a – b)(a2 + ab + b2)
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2

4. Check: Multiplying the factors.
Factoring a Polynomial
1. GCF first factor out any monomial GCF
2. Consider Number of Terms
3. 4 terms: Factor by grouping
4. 3 terms:
a) perfect square: a2 + 2ab + b2 = (a + b)2 Or a2  2ab + b2 = (a  b)2.
b) Not a perfect square:
c) x2 + bx + c, find two factors of c whose sum = b & (x + 1st # )(x + 2nd # )
d) ax2 + bx + c, where a  1,
Use trial and error. Or,
find 2 factors of ac whose sum = b; write these factors as coefficients of two like terms that, when combined, equal bx; and then factor by grouping.
5. 2 terms:
a) A difference of squares: a2 – b2 = (a + b)(a – b)
b) A sum of cubes: a3 + b3 = (a + b)(a2 – ab + b2).
c) A difference of cubes: a3 – b3 = (a – b)(a2 + ab + b2).
Check: Multiplying the factors.

5. Examples 1: Factor. a.) 2x4 – 50x2 b.) 24x2 – 6xy + 40xy – 10y2
a2 – b2 = (a + b)(a – b)
a3 + b3 = (a + b)(a2 – ab + b2)
a3 – b3 = (a – b)(a2 + ab + b2)
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
Factor.
a.) 2x4 – 50x2
b.) 24x2 – 6xy + 40xy – 10y2
c.) 12x2 – 8x – 15
= (6x + 5)(2x – 3)

6. Examples 2: Factor. a) 5x3 – 10x2 – 120x b) 12y5 + 84y3 c) 8a4 – 72n2
a2 – b2 = (a + b)(a – b)
a3 + b3 = (a + b)(a2 – ab + b2)
a3 – b3 = (a – b)(a2 + ab + b2)
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
Examples 2:
Factor.
a) 5x3 – 10x2 – 120x b) 12y5 + 84y3 c) 8a4 – 72n2
Solution
= 5x(x2 – 2x – 24)
= 5x(x + 4)(x – 6)
= 8(a4 – 9n2)
= 12y3(y2 + 7)
= 8(a2 – 3n)(a2 + 3n)

7. Examples 3: Factor a)150x3y – 120x2y2 + 24xy3 Solution
a2 – b2 = (a + b)(a – b)
a3 + b3 = (a + b)(a2 – ab + b2)
a3 – b3 = (a – b)(a2 + ab + b2)
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
a)150x3y – 120x2y2 + 24xy3
Solution
6xy(25x2 – 20xy + 4y2 )
= 6xy(5x – 2y)2
b)x5 – 2x3 – 27x2 + 54
= x3(x2 – 2) – 27(x2 – 2)
= (x2 – 2)(x3 – 27)
= (x2 – 2)(x – 3)(x2 + 3x + 9)

8. Examples 4:
a2 – b2 = (a + b)(a – b)
a3 + b3 = (a + b)(a2 – ab + b2)
a3 – b3 = (a – b)(a2 + ab + b2)
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
Factor
a)100x2 – 60xy + 9y2 b) x3 – 1 8 y3 c) 3x2 – 18x + 27 – 3y2