VertexMath · MPM2D · Free worksheet

Factoring Algebraic Expressions Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 45 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Core Skills (K)

GCF & Grouping · Simple Trinomials · Decomposition · Special Cases · Factoring Completely

K1 Common Factors & Grouping

Factor.

a)
$14a^3b^2-21a^2b^4+7a^2b^2$
b)
$x^3-4x^2-3x+12$
c)
$6xy+15-10y-9x$

Stuck? Peek at a hint

Use the largest number and the lowest power of each variable that divide every term. For four terms, group them in pairs that share a factor; you may reorder the terms.

K2 Simple Trinomials

Factor.

a)
$x^2+11x+24$
b)
$x^2-2x-35$
c)
$x^2-13xy+36y^2$

Stuck? Peek at a hint

List the factor pairs of the constant term, with their signs, and pick the pair whose sum is the middle coefficient.

K3 Complex Trinomials

Factor completely.

a)
$3x^2+10x+8$
b)
$6x^2-7x-20$
c)
$20x^2-26x+6$

Stuck? Peek at a hint

Remove any common factor first. Then find two numbers with product $ac$ and sum $b$ , and use them to split the middle term.

K4 Special Cases?

Factor. Not every expression is a special case.

a)
$49x^2-36y^2$
b)
$4x^2-20x+25$
c)
$9x^2+30x+16$

Stuck? Peek at a hint

Take the square roots of the first and last terms, then test whether twice their product equals the middle term.

K5 Factor Completely

Factor completely. Each expression needs more than one step.

a)
$-2x^2+14x-20$
b)
$16x^4-1$
c)
$x^3+2x^2-9x-18$

Stuck? Peek at a hint

Start with the GCF (a negative one if the leading coefficient is negative) or with grouping. After each step, test every bracket again.

Real World (A)

Revenue · Area · Volume

A1 Bike Rentals

A bike-rental shop in Toronto models its daily revenue as $R=-6x^2-10x+1200$ dollars, where $x$ is the number of $2 increases to its daily price of $30 per bike.

a)
Factor $R$ completely.
b)
Rewrite $R$ as a product of two factors, one of which is the price per bike, $30+2x$ . What does the other factor represent? How many bikes are rented at $30?
c)
The factored form gives $R=0$ at $x=\frac{40}{3}$ . Can the shop set this price? Find the largest number of increases for which at least one bike is rented, and the revenue then.

Stuck? Peek at a hint

Take out the common factor $-2$ first, then decompose the trinomial that is left.

A2 Plaza Paving

A square plaza in Kingston has side length $s$ metres. A square fountain of side $f$ metres sits in the middle, and the rest of the plaza is paved.

a)
Write the paved area in factored form.
b)
The plaza is $46$ m wide and the fountain is $14$ m wide. Paving costs $85 per m². Use your factored form to find the paving cost.
c)
A designer sketches a fountain $50$ m wide for the same plaza. Evaluate your factored form and say what the result tells the designer.

Stuck? Peek at a hint

Paved area = plaza area − fountain area.

A3 Cardboard Tray

An open-top tray is made from a $30$ cm by $20$ cm sheet of cardboard: a square of side $x$ cm is cut from each corner, then the sides are folded up. The tray's volume is $V=4x^3-100x^2+600x$ cm³.

a)
Factor $V$ completely.
b)
Rewrite your answer as height × width × length of the tray. Then find the volume when $x=4$ .
c)
The factored form also gives $V=0$ at $x=15$ . Can 15 cm squares be cut from this sheet? State the realistic values of $x$ .

Stuck? Peek at a hint

Start by removing the common factor $4x$ .

Think It Through (T)

Error Analysis · Counterexamples

T1 Spot the Flaw

A student factors $2x^2-5x-3$ :
" $ac=-6$ . The numbers $-6$ and $1$ multiply to $-6$ and add to $-5$ , so $2x^2-5x-3=(x-6)(x+1)$ ."
The graph of $y=2x^2-5x-3$ is shown. Each grid square is 1 unit.

a)
Use the graph to show that the student's answer is wrong.
b)
Identify the student's error, then factor $2x^2-5x-3$ correctly.
c)
The student's shortcut does work for $x^2-5x-6$ . Explain why it works there but not for $2x^2-5x-3$ .

Stuck? Peek at a hint

Read the $x$ -intercepts from the graph and compare them with the zeros of $(x-6)(x+1)$ .

T2 Test the Claim

A student claims $(x+3)^2=x^2+9$ and tests it with a table.

$x$	$0$	$1$	$-3$
$(x+3)^2$	$9$
$x^2+9$	$9$

a)
The student tested only $x=0$ . Complete the table. Does it support the claim?
b)
Expand $(x+3)^2$ . For which values of $x$ are the two expressions equal?
c)
A second student claims $x^2+9=(x+3)(x-3)$ . Give a counterexample, then explain why $x^2+9$ cannot be factored over the integers.

Stuck? Peek at a hint

Substitute the same value of $x$ into both sides of each claim.

Boss Level

Optional challenge

Differences of Squares

Some whole numbers can be written as a difference of two perfect squares, for example $40=7^2-3^2$ .

a)
Find positive integers $m$ and $n$ with $m^2-n^2=96$ and $m-n=2$ .
b)
Find all pairs of positive integers $(m,n)$ with $m^2-n^2=96$ .
c)
Explain why no integers $m$ and $n$ satisfy $m^2-n^2=98$ .

Stuck? Peek at a hint

Factor the left side as $(m-n)(m+n)$ , then look at the factor pairs of $96$ .

Answer key

Final answers only. Try first, then check.

Core Skills

a) $7a^2b^2(2a-3b^2+1)$
b) $(x-4)(x^2-3)$
c) $(3x-5)(2y-3)$

a) $(x+3)(x+8)$
b) $(x-7)(x+5)$
c) $(x-4y)(x-9y)$

a) $(3x+4)(x+2)$
b) $(3x+4)(2x-5)$
c) $2(10x-3)(x-1)$

a) $(7x-6y)(7x+6y)$
b) $(2x-5)^2$
c) $(3x+2)(3x+8)$

a) $-2(x-2)(x-5)$
b) $(4x^2+1)(2x-1)(2x+1)$
c) $(x+2)(x-3)(x+3)$

Real World

a) $R=-2(3x-40)(x+15)$
b) $R=(30+2x)(40-3x)$ ; $40-3x$ is the number of bikes rented per day; $40$ bikes at $30.
c) No: $\frac{40}{3}$ is not a whole number of $2 increases. Largest: $x=13$ (price $56, $1$ bike), so $R=\$56$ .

a) $(s-f)(s+f)$ m²
b) $(32)(60)=1920$ m², so $163,200
c) $(46-50)(46+50)=-384$ : a negative area is impossible. The fountain is wider than the plaza; it must be less than $46$ m wide.

a) $V=4x(x-10)(x-15)$
b) $V=x(20-2x)(30-2x)$ ; at $x=4$ : $4\times12\times22=1056$ cm³
c) No: the sheet is only $20$ cm wide, so the width $20-2x$ would be $-10$ cm. Realistic values: $0<x<10$ .

Think It Through

a) $(x-6)(x+1)$ is zero at $x=6$ and $x=-1$ , but the graph crosses the $x$ -axis at $x=-\frac{1}{2}$ and $x=3$ (and its $y$ -intercept is $-3$ , not $-6$ ).
b) $-6$ and $1$ split the middle term; they do not go in the brackets. $2x^2-6x+x-3=2x(x-3)+1(x-3)=(2x+1)(x-3)$
c) When $a=1$ , $ac=c$ and grouping gives $x(x-6)+1(x-6)=(x+1)(x-6)$ , so the numbers land in the brackets. When $a\ne1$ , grouping puts the factor $2$ into a bracket: $(2x+1)(x-3)$ .

a) $x=1$ : $16$ and $10$ ; $x=-3$ : $0$ and $18$ . No: one counterexample ( $x=1$ ) disproves the claim.
b) $x^2+6x+9$ ; equal only at $x=0$ (they differ by $6x$ ).
c) Any $x$ works, e.g. $x=0$ : $9\ne-9$ , since $(x+3)(x-3)=x^2-9$ . Two integers with product $9$ have the same sign, so their sum cannot be $0$ .

Boss Level

Boss

a) $m=25$ , $n=23$
b) $(25,23)$ , $(14,10)$ , $(11,5)$ , $(10,2)$
c) $m-n$ and $m+n$ differ by $2n$ , so both are odd or both are even. Their product is then odd or a multiple of $4$ , and $98$ is neither.

Free cheat sheet

Factoring Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

Unit Summary

One printable page with every rule.

↓ Unit Summary PDF

Build your own

The same page with blanks: fill it in from memory, then check.

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Phone card

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↓ Phone card (image)

Most testedFactor completely: $6x^2+3x-18$ . (card 4)

1Common factor (GCF)

UseTake out the largest number and the lowest power of each shared variable. Divide every term by it.

Example $12x^3y-18x^2y^2+6x^2y=6x^2y(2x-3y+1)$ .

Trap $5a^2+5a=5a(a+1)$ : the term equal to the GCF leaves $1$ , not nothing.

TrapThe GCF stays in the answer: $4x^2-12x=4x(x-3)$ , never $(x-3)$ .

2Grouping

UseFour terms: group in pairs, take the GCF of each pair, then take out the common bracket. If the brackets don't match, try another pairing.

Example $x^3+5x^2-2x-10=x^2(x+5)-2(x+5)=(x+5)(x^2-2)$ .

TrapTaking out a negative flips both signs: $-3x+12=-3(x-4)$ , not $-3(x+12)$ .

3 $x^2+bx+c$

$x^2+bx+c=(x+m)(x+n)$

$mn=c\qquad m+n=b$

$c>0$ , $b>0$	both $+$
$c>0$ , $b<0$	both $-$
$c<0$	opposite signs

Example $x^2-3x-28$ : $(-7)(4)=-28$ and $-7+4=-3$ , so $(x-7)(x+4)$ .

TipWhen $c<0$ , the number with the larger size has the sign of $b$ .

Trap $x^2-4x-21=(x-7)(x+3)$ , not $(x-7)(x-3)$ : $c<0$ needs opposite signs.

4 $ax^2+bx+c$ : decomposition

UseGCF first. Find $m$ , $n$ with $mn=ac$ and $m+n=b$ . Split $bx$ into $mx+nx$ , then group.

Example $2x^2+7x+3$ : $ac=6$ and $6+1=7$ , so $2x^2+6x+x+3=(2x+1)(x+3)$ .

Trap $6$ and $1$ split the middle term. They don't go into brackets: $(x+6)(x+1)$ is wrong.

Trap $3x(x+2)-4x-8$ : take out $-4$ , giving $-4(x+2)$ , so both brackets match.

5Special products

$a^2-b^2=(a-b)(a+b)$

$a^2\pm2ab+b^2=(a\pm b)^2$

UseSquare-root the coefficients: $25x^2-16=(5x-4)(5x+4)$ . A sum of squares $a^2+b^2$ does not factor.

Trap $4x^2+13x+9$ is not $(2x+3)^2$ : the middle term would be $12x$ . Check $2ab$ first.

6Factor completely

Use1. GCF first (negative if the first term is). 2. Count terms: two, difference of squares; three, trinomial; four, grouping. 3. Try to factor every bracket again.

Example $3x^3-12x=3x(x^2-4)=3x(x-2)(x+2)$ .

TipExpand your answer to check it.

Trap $x^4-81=(x^2+9)(x^2-9)$ is not finished: $(x^2+9)(x-3)(x+3)$ .

TrapTaking out $-2$ flips every sign: $-2x^2+6x+8=-2(x^2-3x-4)$ .

After the GCF

Two squares, subtracted

(a-b)(a+b)

Three terms,

x^2

alone Product and sum

Three terms,

ax^2

with

a\ne1

Decomposition

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Core Skills (K)

K1 Common Factors & Grouping

K2 Simple Trinomials

K3 Complex Trinomials

K4 Special Cases?

K5 Factor Completely

Real World (A)

A1 Bike Rentals

A2 Plaza Paving

A3 Cardboard Tray

Think It Through (T)

T1 Spot the Flaw

T2 Test the Claim

Boss Level

Differences of Squares

Answer key

Core Skills

Real World

Think It Through

Boss Level

Factoring Cheat Sheet

Unit Summary

Build your own

Phone card

1Common factor (GCF)

2Grouping

3x2+bx+cx^2+bx+cx2+bx+c

4ax2+bx+cax^2+bx+cax2+bx+c: decomposition

5Special products

6Factor completely

After the GCF

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3 $x^2+bx+c$

4 $ax^2+bx+c$ : decomposition