Factoring is finding common factors (usually the GCF) between all the terms in a long algebraic expression and using them to rewrite the expression as a product of many factors. In some ways it can be thought of as the opposite of expanding brackets as factoring puts many terms into products of brackets. We've already come across some of the basics of factoring in Adding in brackets and Letters as factors.
To factor an expression, just follow the following format for a three-term expression:
$AB+AC+AD=A\left(B+C+D\right)$AB+AC+AD=A(B+C+D)
where $A$A is the GCF of all the terms, and can be extended to examples with more or less than three terms.
Where there are many different variables involved, just look at each one individually and see what the GCF is between the terms.
Factor the following expression by taking out the highest common factor:
$7x+35$7x+35
Factor the following expression by taking out the highest common factor:
$42x-x^2$42x−x2
Factor the following by taking out the highest negative common factor: $-5x^2+20x$−5x2+20x
Think: about how the signs will change after taking out a negative factor
Do:The negative GCF between the coefficients $-5$−5 and $20$20 is $-5$−5. The GCF between $x^2$x2 and $x$x is $x$x.
Therefore our overall negative GCF is $-5x$−5x.
The first term $-5x^2$−5x2 will be positive after dividing by $-5x$−5x, and $20x$20x will be negative after dividing by $-5x$−5x.
So $-5x^2+20x=-5x\left(x-4\right)$−5x2+20x=−5x(x−4)
$4xy^2+2xy$4xy2+2xy | $=$= | $\left(2xy\right)\left(2y\right)+\left(2xy\right)\times\left(1\right)$(2xy)(2y)+(2xy)×(1) | |
Greatest Common Factor is $2xy$2xy, so remove that from each component | |||
$=$= | $2xy\left(2y+1\right)$2xy(2y+1) | Factoring completely |
Factor the following expression:
$2t^2k^7+18t^9k^9$2t2k7+18t9k9
Factor binomials (e.g., 4x^2 + 8x) and trinomials (e.g., 3x^2 + 9x – 15) involving one variable up to degree two, by determining a common factor using a variety of tools and strategies (e.g., patterning)