6.4 Factoring and Solving Polynomial

Equations

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)

Review

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)

Special Factoring Patterns

Factor each polynomial. x3 + 27

8x3 - 125

Examples

Factor x3 + 343

Your Turn!

You can sometimes factor out a monomial, then use sum/difference of cubes.

(When powers are 3 apart.) Example:

◦64x4 – 27x

Factoring a Common Monomial First

Factor 3x4 – 24x

Your Turn!

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

Factoring by Grouping

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

Example:

Factor x3 – 2x2 +4x - 8

Your Turn!

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

Factoring in Quadratic Form

Factor 4x6 – 20x4 + 24x2

Factor x4 + 4x2 - 21

Examples

Factor x4 + 3x2 + 2

Your Turn!

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.)

Solving Polynomial Equations by Factoring

Find the real number solutions of the equation.

2x5 + 24x = 14x3

2x5 – 18x = 0

Example

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?

Real-Life Applications

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?

Your Turn!