Factorization

Ontario 10 Academic (MPM2D)

Simplify algebraic fractions

Lesson

An interesting application of factoring is helping to simplify fractions, as both processes involve looking at factors. Let's have a quick review of how to do exactly that with algebraic fractions.

Simplifying fractions already taught us how to simplify fractions that only involve numbers, and algebraic fractions work exactly the same way. Simply find common factors (greatest common factor [GCF] is the fastest way) between the denominator and the numerator and cancel them out until you can't find any more.

For example, $\frac{49st^2}{42t}$49`s``t`242`t` involves numbers and the two variables $s$`s` and $t$`t`, so let's look at them all separately. Just looking at the numbers, we see that $7$7 is the $GCF$`G``C``F` between $49$49 and $42$42, so we can take it out from both to leave $7$7 and $6$6, respectively. As for $s$`s` terms, there are no common factors between the denominator and numerator except $1$1 so we leave them. Lastly the $t$`t` terms have a GCF of $t$`t`, so we are left with $t$`t` on the top and $1$1 on the bottom after cancelling out. So in the end we should have $\frac{7st}{6}$7`s``t`6 after simplifying.

As previously mentioned, factoring can greatly help us simplify complicated algebraic fractions but we can use all the methods we've learnt so far to similarly simplify fractions. Have a look at the following examples to see what I'm talking about.

Factor and simplify $\frac{x^2-6x+9}{x-3}$`x`2−6`x`+9`x`−3

**Think **about whether the numerator can be factored using quadratic methods or perfect square methods

**Do**

$x^2-6x+9$`x`2−6`x`+9 can be factored using the perfect square method as $9$9 is a square number, and $6$6 is double $\sqrt{9}=3$√9=3

Therefore it becomes $\left(x-3\right)^2$(`x`−3)2

So our fraction can be rewritten as:

$\frac{\left(x-3\right)^2}{x-3}=x-3$(`x`−3)2`x`−3=`x`−3 as $x-3$`x`−3 is a common factor of the numerator and denominator

Factor and simplify $\frac{y+4}{y^2-3y-28}$`y`+4`y`2−3`y`−28

**Think **about which method to use for the denominator and how the negative $-28$−28 will affect it

**Do**

$y^2-3y-28$`y`2−3`y`−28 is a monic quadratic trinomial but not a perfect square as $-28$−28 is not a square number

Its negativity also means the two numbers $a$`a` and $b$`b` we need to find in $\left(y+a\right)\left(y+b\right)$(`y`+`a`)(`y`+`b`) have different signs.

Number pairs that give us $-28$−28 are:

$1$1 & $-28$−28, $-1$−1 & $28$28, $2$2 & $-14$−14, $-2$−2 & $14$14, $4$4 & $-7$−7, $-4$−4 & $7$7

The only pair to have a sum of $-3$−3 is $4$4 & $-7$−7, which must be our $a$`a` and $b$`b`

$\left(y+4\right)\left(y-7\right)$(`y`+4)(`y`−7) must then be the factored form

Our fraction then becomes $\frac{y+4}{\left(y+4\right)\left(y-7\right)}=\frac{1}{y-7}$`y`+4(`y`+4)(`y`−7)=1`y`−7 as we take the $y+4$`y`+4 out

Factor and simplify $\frac{jk-j+k^2-k}{k\left(j+k\right)}$`j``k`−`j`+`k`2−`k``k`(`j`+`k`)

**Think** about which method applied to the four termed numerator

**Do**

$jk-j+k^2-k$`j``k`−`j`+`k`2−`k` can be factored by grouping in pairs, so it becomes

$j\left(k-1\right)+k\left(k-1\right)$`j`(`k`−1)+`k`(`k`−1) = $\left(k-1\right)\left(j+k\right)$(`k`−1)(`j`+`k`)

So the fraction becomes

$\frac{\left(k-1\right)\left(j+k\right)}{k\left(j+k\right)}=\frac{k-1}{k}$(`k`−1)(`j`+`k`)`k`(`j`+`k`)=`k`−1`k` if you take the $j+k$`j`+`k` out

Factor $\frac{6x-16}{12}$6`x`−1612 and simplify.

Factor and simplify $\frac{50m^2+70mn}{80m^2}$50`m`2+70`m``n`80`m`2.

Factor and simplify $\frac{a^2-81}{9-a}$`a`2−819−`a`.

Factor polynomial expressions involving common factors, trinomials, and differences of squares, using a variety of tools and strategies