Factor and Solve Polynomial Equations


Factor and Solve Polynomial Equations

Goal: Solving polynomial equations by factoring.


Polynomial Identities

In order to solve polynomials we need to be aware of basic identities which allow us to simplify the equation. 


The following are the main identities for binomial factors


1.) x2 − y2 = (x − y)(x + y)
Example:  9 − p2  =  (3 − p)(3 + p)

2.) x3 + y3 = (x + y)(x2 − xy + y2)
Example:  64 + 8x3  =  (4 + 2x)(16 − 8x  + 4x2)

3.) x3 − y3 = (x − y)(x2 + xy + y2)
Example:  64 − 8x3  =  (4 − 2x)(16 + 8x + 4x2)

4.) x4 − y4 = (x − y)(x + y)(x2 + y2)
Example:  81y4 − 625x8  =  (3y − 5x2)(3y + 5x2)(9y2 + 25x4)


Solve by Factoring

Example

Solve for x in the equation by factoring and find all the complex roots:  2x3 + 16 = 0


Solution

Since 2 is a common factor to each term, factor out 2:

2(x3 + 8) = 0

Then factor the remaining cubic expression:

2(x + 2)(x2 − 2x + 4) = 0

Use the zero-product property:

2 = 0   OR   x + 2 = 0   OR   x2 − 2x + 4 = 0


Solve each equation:

2 = 0

2 ≠ 0, so this one gets discarded

x + 2 = 0

x = −2

x2 − 2x + 4 = 0

x =

=


and we find the solutions

x = {−2,  1 + ,  1 − }




Practice

Find the solutions to the following equations.

1.  3x2 − 8 = 100

2.  x3 − 100 = 25

3.  x4 − 81 = 0


















Answers

1.  x = {-6, 6}

2.  x = {5, , }

3.  x = {-3, 3, -3i, 3i}