Explore: Factoring to Solve Solutions

Factoring to Solve Solutions

One method that can be used to solve polynomial expressions is factoring . It is not possible to solve every polynomial by factoring, but it is a method that can be used in many cases. 
To factor:

1. Find the greatest common factor, GCF , (other than one). 
2. Factor by grouping when given four terms. 
3. Factor special products (difference of two squares, sum/difference of cubes, perfect square trinomials). 
4. Solve using your preferred factoring method (or another more efficient solving method).

Note

These steps are slightly different than those presented in Algebra 1. Methods other than factoring are used to solve in this unit.

Once the expression is factored, solve using the Zero-Product Property . 
The Zero-Product Property: 
- Says: If $a b = 0$ , then $a = 0$ , or $b = 0$ .
-  Allows you to solve an equation by setting each expression equal to zero and solving for the value(s) of the variable. 
Remember that the degree of the equation tells you the number of solutions, or roots .  
You will also use the Conjugate Root Theorem to factor equations and then solve.

Example 6

Solve.

$x^{2} + 2 x \sqrt{3} + 3 = 0$

Implement

$x^{2} + 2 x \sqrt{3} + 3 = 0 (x +) (x +) = 0$

$(x + \sqrt{3}) (x + \sqrt{3}) = 0 x = - \sqrt{3}$

Explain

Special product pattern: Perfect Square Trinomial

The Conjugate Root Theorem does not apply to this problem because the middle coefficient is not a rational number. This is an example of a repeated root.

Example 7

Solve.

$4 x^{3} + 8 x^{2} - 5 x - 10 = 0$

Implement

$(4 x^{3} + 8 x^{2}) + (- 5 x - 10) = 0$

$4 x^{2} (x + 2) - 5 (x + 2) = 0 (4 x^{2} - 5) (x + 2) = 0$

$(2 x + \sqrt{5}) (2 x - \sqrt{5}) (x + 2) = 0$

$x = \pm \frac{\sqrt{5}}{2}, - 2$

Explain

Factor by grouping
Recall: $x^{2} - d = (x + \sqrt{d}) (x - \sqrt{d})$
Zero-Product Property

The term with the ± symbol will be written first so as not to confuse this with a single expression for the solution.

Example 8

Solve.

$x^{2} = - 5 x$

Implement

$x^{2} + 5 x = 0 x (x + 5) = 0 x = 0, - 5$

Explain

Set equal to zero
 GCF

Watch out! When problems are written in $a x^{2} + b x = 0$ form, factor out the GCF to solve.