The document provides examples and instructions for factoring polynomials of various types, including: - Trinomials like x^2 - 5x - 12 - Sum and difference of cubes like x^3 + 8 and 8x^3 - 1 - Polynomials with a common monomial factor like 6x^2 + 15x - Quadratics in the form of au^2 + bu + c It also discusses using the zero product property to solve polynomial equations by factoring and setting each factor equal to zero.

2.  We already know how to factor these
types of quadratic expressions:
◦ General Trinomial:
2x2 – 5x – 12 = (2x + 3)(x – 4)
◦ Perfect Square Trinomial:
x2 + 10x + 25 = (x + 5)2
◦ Difference of Two Squares:
4x2 – 9 = (2x + 3)(2x – 3)
◦ Common Monomial Factor:
6x2 + 15x = 3x(2x + 5)

3.  Sum of Two Cubes:
a3 + b3 = (a + b)(a2 – ab + b2)
 Example:
x3 + 8 = (x + 2)(x2 – 2x + 4)
 Difference of Two Cubes:
a3 – b3 = (a – b)(a2 + ab + b2)
 Example:
8x3 – 1 = (2x – 1)(4x2 + 2x + 1)

4.  Factor each polynomial.
 x3 + 27
 8x3 - 125

5.  Factor x3 + 343

6.  You can sometimes factor out a
monomial, then use sum/difference of
cubes.
 (When powers are 3 apart.)
 Example:
◦ 64x4 – 27x

7.  Factor 3x4 – 24x

8.  Sometimes we can factor by grouping
pairs of terms with a common monomial
factor:
ra + rb + sa + sb = r(a + b) + s(a + b)
= (r + s)(a + b)
 Example:
x3 – 2x2 – 9x + 18

9.  Factor the polynomial.
x2y2 – 3x2 – 4y2 + 12

10.  Factor x3 – 2x2 +4x - 8

11.  An expression of the form
au2 + bu + c where u is any expression
in x is in quadratic form.
 We can factor these like a quadratic.
 Example:
Factor 81x4 - 16

12.  Factor 4x6 – 20x4 + 24x2
 Factor x4 + 4x2 - 21

13.  Factor x4 + 3x2 + 2

14.  Remember the zero product property?
 If AB = 0 then A = 0 or B = 0
 To solve, factor and set each factor equal
to zero. Then solve for x.
 Only include real solutions.
 May have up to n solutions. (where n is
the degree.)

15.  Find the real number solutions of the
equation.
 2x5 + 24x = 14x3
 2x5 – 18x = 0

16.  A large concrete block is discovered by
archaeologists with a volume of 330 cubic
yards. The dimensions are x yards high
by 13x – 11 yards long by 13x – 5 yards
wide. What is the height?

17.  You are building a bin to hold mulch for
your garden. The bin will hold 162 cubic
feet of mulch. The dimensions are x ft
by 5x – 6 ft by 5x – 9 ft. How tall will the
bin be?