Previously, we looked at how to use the distributive law to remove parentheses from algebraic expressions. Now we are going to look at this process in reverse so we can factor expressions (which means writing them with parentheses).
To factorize algebraic equations, we need to find the greatest common factors (GCF) between the terms. We need to consider both the numeric values and algebraic terms. We will start by looking at algebraic factors.
There are lots of ways to look for the greatest common factor between two numbers such as looking at the prime factorization or listing all factors. With lots of practice, you will be able to identify the GCF more quickly.
Factor: $12x+20$12x+20
Think: The $x$x factor is not common between $12x$12x and $20$20, so we are looking for a numeric common factor. We need to consider factors of 12 and 20 to find the greatest common factor between $12x$12x and $20$20 is $4$4.
Factors of $12$12: $1,2,3,4,6,12$1,2,3,4,6,12
Factors of $20$20: $1,2,4,5,10,20$1,2,4,5,10,20
With the GCF of $4$4, we need to break $12x$12x and $20$20 into a product with $4$4.
$12x=4\left(3x\right)$12x=4(3x)
$20=4(5)$20=4(5)
Do:
$12x+20=4\left(3x+5\right)$12x+20=4(3x+5)
Reflect: We can always check to see if we have factored correctly by distributing it back out again.
$4\left(3x+5\right)$4(3x+5) | $=$= | $4\times3x+4\times5$4×3x+4×5 | |
$=$= | $12x+20$12x+20 | We got back the original expression, so we are correct! |
Remember to consider how the operators will change if negative numbers are involved.
Factor: $-100r-20$−100r−20
Think: There is no factor of r in the second term, so the GCF between $-100r$−100r and $-20$−20 must be numeric. $-20$−20 is a factor of $-100$−100, so the GCF is $-20$−20.
$-100r=-20\times5r$−100r=−20×5r
$-20=-20\times1$−20=−20×1
Do: $-20\left(5r+1\right)$−20(5r+1)
Reflect: When the common factor is the same as one of the terms, we can't forget to put the factor of $1$1 in the parentheses. We can always check to see if we have factored correctly by distributing it back out again.
$-20\left(5r+1\right)$−20(5r+1) | $=$= | $-20\times5r-20\times1$−20×5r−20×1 | |
$=$= | $-100r-20$−100r−20 | We got back the original expression, so we are correct! |
Factor the expression $6v+30$6v+30.
Factor the expression $-8v^2+56$−8v2+56.
In addition to there being a numeric common factor, there may also be an algebraic common factor. In order to find the greatest common algebraic factor we can use the following steps.
What is the greatest common algebraic factor of $2x^2y$2x2y and $6xz$6xz?
Think: We need to look at the numeric coefficient first, then the algebraic variables, and then combine our findings into a solution.
Do: The numerical coefficients are $2$2 and $6$6, so the numerical GCF is $2$2.
The algebraic terms present in the first term are $x$x and $y$y, and those present in the second term are $x$x and $z$z. So $x$x is the only common variable between the two. The least power of $x$x present in both terms is $x^1=x$x1=x, so this will be the algebraic component of our greatest common algebraic factor.
Reflect: Our greatest common algebraic factor is $2x$2x.
The same process applies if we are asked to compare $3$3 or more terms, we just need to ensure any common numeric or algebraic variables are common to all terms.
What is the greatest common algebraic factor of the terms $9ax$9ax and $8ba$8ba?
Find the greatest common factor of $x^2y^4+x^5y^6$x2y4+x5y6.
We've already learned how to factor (or add parentheses to) expressions, but only when these expressions have common factors that are numbers. Now we are going to look at expressions that have both numbers and algebraic terms as factors.
Factor: $a^2-3a$a2−3a
Think: There is no common numerical factor. There is a common algebraic factor. The greatest algebraic factor is $a$a. So the GCF is $a$a.
$a^2=a\left(a\right)$a2=a(a)
$-3a=-3\left(a\right)$−3a=−3(a)
Do:
$a^2-3a=a\left(a-3\right)$a2−3a=a(a−3)
Factor: $18w^4-27w^2$18w4−27w2
Think: The greatest common numerical factor is $9$9. The greatest common algebraic factor is $w^2$w2. So the GCF is $9w^2$9w2.
$18w^4=9w^2\left(2w^2\right)$18w4=9w2(2w2)
$-27w^2=9w^2\left(-3\right)$−27w2=9w2(−3)
Do:
$18w^4-27w^2=9w^2\left(2w^2-3\right)$18w4−27w2=9w2(2w2−3)
Factor: $2u^2-8u$2u2−8u
Factor the following expression:
$z^2+4z^5$z2+4z5
Factor the following expression by taking out the greatest common factor:
$9x^2y^2z-18xyz$9x2y2z−18xyz
Factor $9r^2t^2v^3+6r^3tv-12rt^3v^2$9r2t2v3+6r3tv−12rt3v2.
Determine the quotient of a polynomial of degree one and polynomial of degree two when divided by a polynomial of degree one and polynomial of degree two when the degree of the divisor does not exceed the degree of the dividend
Rewrite polynomial expressions of degree one and degree two in equivalent forms using the distributive property