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.)**

*x*^{2}−*y*^{2}= (*x*−*y*)(*x*+*y*)**Example**

**:**9 −

*p*

^{2}= (3 −

*p*)(3 +

*p*)

**2.)**

*x*^{3}+*y*^{3}= (*x*+*y*)(*x*^{2}**−**

*xy*+*y*^{2})**Example:**64 + 8

*x*

^{3}= (4 + 2

*x*)(16 − 8

*x*+ 4

*x*

^{2})

**3.)**

*x*^{3}−*y*^{3}= (*x*−*y*)(*x*^{2}+*xy*+*y*^{2})**Example:**64 − 8

*x*

^{3}= (4 − 2

*x*)(16 + 8

*x*+ 4

*x*

^{2})

**4.)**

**x**

^{4}−*y*^{4}= (*x*−*y*)(*x*+*y*)(*x*^{2}+*y*^{2})**Example**

**:**81

*y*

^{4}− 625

*x*

^{8}= (3

*y*− 5

*x*

^{2})(3

*y*+ 5

*x*

^{2})(9

*y*

^{2}+ 25

*x*

^{4})

**Solve by Factoring**

**Example**

Solve for *x* in the equation by factoring and find all the complex roots: **2 x^{3} + 16 = 0**

**Solution**

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

2(*x*^{3} + 8) = 0

Then factor the remaining cubic expression:

2(*x* + 2)(*x*^{2} − 2*x* + 4) = 0

Use the zero-product property:

2 = 0 OR *x* + 2 = 0 OR *x*^{2} − 2*x* + 4 = 0

Solve each equation:

2 ≠ 0, so this one gets discarded

*x* = 0 − 2 = −2

and we find the solutions