VertexMath · MCR3U · Free worksheet

Polynomials and Rational Expressions Review

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

Core path ~55 min Full sheet ~1 h 50 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)

Polynomials · Factoring · Simplifying · Multiplying & Dividing · Adding & Subtracting

K1 Expand and Simplify

Expand and simplify.

a)
$(3x^2-5x+2)-(x^2-4x-6)$
b)
$5x-2x(x-4)$
c)
$(2x-3)^2-(x+2)(x-2)$

Stuck? Peek at a hint

Expand every product first, keeping it in brackets, then remove the brackets that follow a minus sign.

K2 Factor Fully

Factor fully.

a)
$-4x^3+12x^2-8x$ , starting with a negative common factor
b)
$6x^2+x-12$
c)
$x^3-3x^2-4x+12$

Stuck? Peek at a hint

Take out the greatest common factor first, then check whether each bracket factors again. With four terms, group them in pairs.

K3 Simplify and Restrict

Simplify each expression and state its restrictions.

a)
$\dfrac{x^2-9}{x^2+x-6}$
b)
$\dfrac{10-2x}{x^2-25}$
c)
$\dfrac{2x^2+5x-3}{4x^2-1}$

Stuck? Peek at a hint

Factor the numerator and the denominator completely, and write the restrictions from the factored denominator before you cancel anything.

K4 Multiply and Divide

Simplify each expression and state its restrictions.

a)
$\dfrac{x^2-4}{3x}\cdot\dfrac{6x^2}{x^2+x-2}$
b)
$\dfrac{x+3}{x-1}\div\dfrac{x^2-9}{x^2-1}$
c)
$\dfrac{2x}{x+4}\div\dfrac{x^2}{x^2-16}\cdot\dfrac{x+1}{x-4}$

Stuck? Peek at a hint

Replace each division by multiplication by the reciprocal, working left to right. List restrictions from every denominator and from each divisor's numerator.

K5 Add and Subtract

Simplify each expression and state its restrictions.

a)
$\dfrac{3x+1}{x-2}-\dfrac{x-5}{x-2}$
b)
$\dfrac{4}{x}+\dfrac{3}{x+2}$
c)
$\dfrac{3x+4}{x^2+x-6}-\dfrac{2}{x-2}$

Stuck? Peek at a hint

Factor each denominator to find the LCD. Put a numerator in brackets before you subtract it.

Real World (A)

Polynomial Models · Rational Models

A1 Open-Top Box

An open-top box is made from a $30$ cm by $20$ cm sheet of card: a square of side $x$ cm is cut from each corner, then the sides are folded up. Let $V(x)$ be the volume of the box in cubic centimetres.

a)
Write $V(x)$ as a product of three factors, expand it into standard form, and find the volume when $x=5$ .
b)
Factor your standard form fully. Which values of $x$ make $V(x)=0$ ?
c)
Which of these zeros, if any, describe a box you could actually fold? State the realistic domain of $V$ .

Stuck? Peek at a hint

Each cut removes $x$ cm from both ends of a side, so the base measures $(30-2x)$ cm by $(20-2x)$ cm.

A2 Paddling the Grand River

A canoeist paddles $6$ km upstream on the Grand River, then $6$ km back downstream. The current flows at $2$ km/h, and the canoeist paddles at $s$ km/h in still water.

a)
Write the total trip time $T(s)$ , in hours, as a single rational expression.
b)
At $s=4$ , find the total time and how much longer the upstream leg takes than the downstream leg.
c)
The expression gives $T(1)=-4$ . What does this mean for the canoeist? State the realistic domain of $T$ .

Stuck? Peek at a hint

Time is distance divided by speed. Upstream, the current slows the canoe to $(s-2)$ km/h.

A3 Token Draw

At a school fair, a bag holds $x$ green tokens and $2$ orange tokens. Two tokens are drawn, one after the other, without putting the first one back. The first is green with probability $\dfrac{x}{x+2}$ . If it is green, the second is green with probability $\dfrac{x-1}{x+1}$ and orange with probability $\dfrac{2}{x+1}$ .

a)
Multiply to write $P(\text{green, green})$ and $P(\text{green, orange})$ as single rational expressions.
b)
Simplify $P(\text{green, green})\div P(\text{green, orange})$ . With $9$ green tokens, how many times as likely is green-green as green-orange?
c)
The simplified ratio gives $-\frac{1}{2}$ when $x=0$ . What does this tell you about a bag with no green tokens?

Stuck? Peek at a hint

To divide by a rational expression, multiply by its reciprocal, then cancel common factors.

Think It Through (T)

Error Analysis · Building Expressions

T1 The Check That Checks Nothing

A student simplifies an expression:
$\frac{x}{x-3}-\frac{x+15}{x^2-9}=\frac{x(x+3)-x+15}{(x-3)(x+3)}=\frac{x^2+2x+15}{(x-3)(x+3)}$
They test both versions in a table and write: "The rows match, so my answer is correct."

$x$	$3$	$-3$
$\text{Original}$	undefined	undefined
$\text{Student's answer}$	undefined	undefined

a)
Find the error in the student's work. Then simplify the expression correctly and state its restrictions.
b)
Explain why the student's table proves nothing.
c)
Choose an allowed value of $x$ and use it to show that the student's answer is wrong and yours is right.

Stuck? Peek at a hint

Expand the numerator yourself, one bracket at a time, and compare it with the student's line.

T2 Reverse-Engineer It

The graph of a rational function $f$ is shown, with its asymptotes dashed and an open circle at $(0,2)$ . Each grid square is 1 unit.

a)
Write an equation for $f$ as a quotient of factored polynomials.
b)
Write a rational expression that simplifies to $\dfrac{2}{x+5}$ and whose restrictions are exactly $x\ne\pm5$ .
c)
Write two different rational expressions that both simplify to $\dfrac{x+1}{x-3}$ but have different restrictions. State the restrictions of each.

Stuck? Peek at a hint

Read the $x$ -intercept and the vertical asymptote first: each gives one linear factor.

Boss Level

Optional challenge

A Sum That Factors

A sum of two squares such as $x^4+4$ can still factor, with one extra idea.

a)
Show that $x^4+4=(x^2+2)^2-(2x)^2$ . Then factor $x^4+4$ into two quadratics.
b)
Find every positive integer $n$ for which $n^4+4$ is prime.
c)
Without a calculator, evaluate $\dfrac{(3^4+4)(7^4+4)}{(1^4+4)(5^4+4)}$ .

Stuck? Peek at a hint

For (b), both quadratic factors are whole numbers, so ask when the smaller one equals $1$ . For (c), factor each bracket with (a) before multiplying anything.

Answer key

Final answers only. Try first, then check.

Core Skills

a) $2x^2-x+8$
b) $-2x^2+13x$
c) $3x^2-12x+13$

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

a) $\dfrac{x-3}{x-2}$ , $x\ne-3,\ 2$
b) $-\dfrac{2}{x+5}$ , $x\ne\pm5$
c) $\dfrac{x+3}{2x+1}$ , $x\ne\pm\frac{1}{2}$

a) $\dfrac{2x(x-2)}{x-1}$ , $x\ne-2,\ 0,\ 1$
b) $\dfrac{x+1}{x-3}$ , $x\ne\pm1,\ \pm3$
c) $\dfrac{2(x+1)}{x}$ , $x\ne0,\ \pm4$

a) $\dfrac{2(x+3)}{x-2}$ , $x\ne2$
b) $\dfrac{7x+8}{x(x+2)}$ , $x\ne-2,\ 0$
c) $\dfrac{1}{x+3}$ , $x\ne-3,\ 2$

Real World

a) $V(x)=x(30-2x)(20-2x)=4x^3-100x^2+600x$ ; $V(5)=1000$ cm³
b) $V(x)=4x(x-10)(x-15)$ ; $x=0$ , $10$ , $15$
c) None: $x=0$ cuts nothing and $x=10$ uses up the whole 20 cm side, so both leave no box; $x=15$ is impossible, since two 15 cm cuts need 30 cm of a 20 cm side. Domain: $0<x<10$ .

a) $T(s)=\dfrac{6}{s-2}+\dfrac{6}{s+2}=\dfrac{12s}{s^2-4}$
b) $T(4)=4$ h; upstream $3$ h, downstream $1$ h, so $2$ h longer
c) Nothing real: at $1$ km/h the canoeist is slower than the current and drifts backwards, so they never reach the turning point and no trip time exists. Domain: $s>2$ .

a) $P(\text{green, green})=\dfrac{x(x-1)}{(x+2)(x+1)}$ ; $P(\text{green, orange})=\dfrac{2x}{(x+2)(x+1)}$
b) $\dfrac{x-1}{2}$ ; $4$ times as likely
c) Nothing: with no green tokens, $P(\text{green, orange})=0$ , so the quotient divides by zero and is undefined. $x=0$ is a restriction that cancelling hid; a ratio of probabilities can never be negative.

Think It Through

a) The minus sign must apply to all of $x+15$ : $x(x+3)-(x+15)=x^2+2x-15$ . Result: $\dfrac{x+5}{x+3}$ , $x\ne\pm3$ .
b) $x=3$ and $x=-3$ are restrictions: any expression with $(x-3)(x+3)$ in its denominator is undefined there, so matching "undefined" says nothing. A test needs allowed values.
c) e.g. $x=0$ : original $=\frac{5}{3}$ , student's answer $=-\frac{5}{3}$ , corrected answer $=\frac{5}{3}$

a) $f(x)=\dfrac{x(x-4)}{x(x-2)}$
b) $\dfrac{2(x-5)}{(x+5)(x-5)}=\dfrac{2x-10}{x^2-25}$
c) e.g. $\dfrac{x(x+1)}{x(x-3)}$ : $x\ne0,\ 3$ ; $\dfrac{(x+1)(x+2)}{(x-3)(x+2)}$ : $x\ne-2,\ 3$

Boss Level

Boss

a) $x^4+4=(x^2-2x+2)(x^2+2x+2)$
b) $n=1$ only ( $1^4+4=5$ ); for $n\ge2$ both factors are greater than $1$
c) $65$

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Most testedSimplify $\dfrac{x^2-9}{x^2+x-6}$ and state the restrictions. (card 4)

1Adding and multiplying

UseCollect like terms: same variables, same exponents. Multiply each term of one bracket by each term of the other.

Example $(3x-2)(x+5)=3x^2+15x-2x-10=3x^2+13x-10$ .

TrapA minus before a bracket flips every sign: $(4x^2-x)-(x^2-3x+2)=3x^2+2x-2$ .

2Special products

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

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

UseThree factors: multiply two of them, then multiply the result by the third.

Trap $(x-2)^2=x^2-4x+4$ , not $x^2-4$ : squaring a binomial keeps the middle term.

3Factoring

UseCommon factor first. $ax^2+bx+c$ : two numbers with product $ac$ and sum $b$ , then group. Four terms: group in pairs. Check by expanding.

Example $2x^2+5x-3$ : $6$ and $-1$ , so $2x^2+6x-x-3=(x+3)(2x-1)$ .

Trap $4x^2-49=(2x-7)(2x+7)$ , not $(4x-7)(4x+7)$ : take the root of the coefficient too.

Trap $-6x^2+9x=-3x(2x-3)$ : divide every term by the negative factor.

4Simplifying

UseFactor fully. State restrictions from the original denominator. Then cancel common factors.

Example $(x^2-9)/(x^2+x-6)=(x-3)(x+3)/[(x+3)(x-2)]=(x-3)/(x-2)$ , $x\ne-3,2$ .

TrapCancel factors, never terms: $(x+5)/(x+7)$ does not simplify to $5/7$ .

5Holes and asymptotes

UseA factor that cancels gives a hole. A factor left in the denominator gives a vertical asymptote.

Example $y=(x-3)/(x-2)$ from card 4: hole at $(-3,\ 6/5)$ , asymptote $x=2$ .

TrapThe simplified graph still has the hole: draw an open circle at $x=-3$ .

6Multiplying and dividing

UseFactor, flip each divisor, then cancel. Restrictions: every original denominator and the numerator of each divisor.

Trap $(2-x)/(x-2)=-1$ , not $1$ : $2-x=-(x-2)$ . Keep $x\ne2$ .

TrapDividing by $(x-4)/(x+1)$ adds $x\ne4$ as well as $x\ne-1$ .

7Adding and subtracting

UseFactor the denominators. LCD: each factor once, at its highest power. Rewrite, combine the numerators in brackets, simplify. Check with an allowed value.

Example $2/x+3/(x+1)=(5x+2)/[x(x+1)]$ , $x\ne0,-1$ .

TrapNever add denominators: $1/x+1/y=(x+y)/(xy)$ , not $2/(x+y)$ .

TrapCheck with a value that isn't restricted: $x=0$ proves nothing here.

Which factoring?

Every term shares a factor Common factor

Two squares, subtracted

(a-b)(a+b)

Three terms Product and sum

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