Choosing Which Factoring Applies
Steps to Factor an Expression
- Check if all terms share a common factor.
- Is each coefficient divisible by the same number?
- Or are there \(x\)s (or \(y\)s or \(z\)s, etc. ) in each term?
- Check if the expression is a difference of squares.
- Are there exactly two terms and are they being subtracted?
- And is each term a perfect square?
- Check if the expression is of the form \(x^2+bx+c\) or \(ax^2+bx+c\).
- Check if the expression is a perfect square in expanded form. Recall that perfect squares can also be factored by Decomposition.
Example 6
Factor.
- \(15x^3y^7+25x^2y^4\)
- \(81x^8-1\)
- \(2x^2+4x-70\)
Solution — Part A
Factor \(15x^3y^7+25x^2y^4\)
The largest common factor is \(5x^2y^4\).
\(15x^3y^7+25x^2y^4 =5x^2y^4(3xy^3+5) \)
The factor that remains once the common factor is removed is not factorable so this is fully factored.
Solution — Part B
Factor \(81x^8-1\)
There are no common factors.
\(81x^8-1\) is a difference of squares since \((9x^4)^2=81x^8\) and \(1^2=1\).
\(81x^8-1 = (9x^4+1)(9x^4-1) \)
Notice that the factor \(9x^4-1\) is also a difference of squares since \((3x^2)^2=9x^4\) and \(1^2=1\). The complete solution is
\(\begin{align*}81x^8-1 & = (9x^4+1)(9x^4-1) \\ & = (9x^4+1)(3x^2-1)(3x^2+1)\end{align*}\)
None of the three factors are factorable; this is fully factored.
Solution — Part C
Factor \(2x^2+4x-70\)
The largest common factor is \(2\).
\(2x^2+4x-70 = 2(x^2+2x-35)\)
Notice that the factor that remains after removing the common factor is of the form \(x^2+bx+c\).
We require two integers
- with sum \(2\), and
- product \(-35\).
The integers are \(7\) and \(-5\).
The full solution is
\(\begin{align*}2x^2+4x-70 & = 2(x^2+2x-35)\\ & =2(x+7)(x-5)\end{align*}\)