Wize High School Grade 10 Math Textbook > Factoring Polynomials

What is Factoring?
Recall
The standard form of a quadratic equation looks like y=ax2+bx+cy=ax^2+bx+cy=ax2+bx+c where a≠0a\neq 0a=0.

It turns out that the standard form equation is not the easiest to work with!
Graphing the standard form of a quadratic equation: 
we need to first create a table of values, plot all of the points, then sketch the curve of best fit
Solving the standard form of a quadratic equation 0=ax2+bx+c\bcth{0=ax^2+bx+c}0=ax2+bx+c:
We would have to guess different values of xxx, substitute them into the equation to see if we can a yyy value of 0

It turns out it's much easier to work with the factored form of a quadratic equation when it comes to graphing and solving quadratic equations!


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The goal of this section is to learn how to use factoring to factor a quadratic equation:

Standard formFactored formy=ax2+bx+cfactoring→y=a(x−r)(x−s)\large{\boxed{\begin{array}{ccccc}
\text{Standard form}&&&&\text{Factored form}\\
y=ax^2+bx+c&&\begin{array}{c}\bcf{\text{factoring}}\\\bcf{\to}\end{array}&&y=a(x-r)(x-s)
\end{array}}}Standard formy=ax2+bx+c​​factoring→​​​Factored formy=a(x−r)(x−s)​​

Wize Tip
Factoring is the opposite of expanding!

Example
Expanding: (x+1)(x−2) → x2−x−2(x+1)(x-2)~\to~x^2-x-2(x+1)(x−2) → x2−x−2
Factoring: x2−x−2 → (x+1)(x−2)x^2-x-2~\to ~(x+1)(x-2)x2−x−2 → (x+1)(x−2)

Once we used factoring to find the factored form of the quadratic, we can always check out answer by expanding the factored form to see if we end up with the original quadratic expression

Different Types of Factoring
Common factoring
Factoring simple trinomials of the form y=x2+bx+cy=x^2+bx+cy=x2+bx+c
Factoring harder trinomials of the form y=ax2+bx+cy=ax^2+bx+cy=ax2+bx+c (a≠1 or 0a\neq 1~\text{or}~0a=1 or 0)
Factoring using difference of squares y=(ax)2−b2y=(ax)^2-b^2y=(ax)2−b2
Factoring perfect square trinomials y=(ax)2+2abx+b2y=(ax)^2+2abx+b^2y=(ax)2+2abx+b2