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Secondary 3 Algebraic Expressions Worksheet

By Marius — private math tutor in Montreal. Updated 30 August 2026

Up to now an algebraic expression was something you tidied up: collect like terms, remove a bracket, substitute a number. Secondary 3 turns the process around. This is the year you learn to put an expression back into a product — first by taking out a common factor, then by grouping four terms in pairs — and the year you multiply two binomials for the first time, which is the operation factoring undoes. Everything on this sheet is the raw material Secondary 4 builds its quadratics out of, whichever option you end up in. Read it here, and print the PDF when you want to write on it; it costs nothing and asks for no account.

Practice worksheet — free PDF

8 pages 12 questions Letter size, print-ready

Download the worksheet Read the questions here

No email, no account, no watermark. Teachers: photocopy it for your classes freely. Worked solutions and 10 harder problems come with the Secondary 3 Math bundle.

All 12 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

Q1Algebra - Algebraic Expressions

A snow-clearing contractor in Sherbrooke charges by the storm. For a storm needing $n$ passes of the plough and $h$ hours of hand shovelling, the charge in dollars is $85 n + 40 h + 120 .$
1. Name the variables, give the constant term, and give the coefficient of $h$ .
2. State the degree of the expression and say whether it is a monomial, a binomial or a trinomial.
3. Find the charge for a storm needing $3$ passes and $2.5$ hours of shovelling.
Q2Dividing an Algebraic Expression by a Monomial

Simplify each quotient.
1. $\frac{45 m^{6} n^{2}}{9 m^{2} n}$
2. $\frac{28 a^{4} b - 12 a^{3} b^{2} + 4 a^{2} b}{4 a^{2} b}$
3. The deck of a skatepark ramp is a rectangle of area $54 p^{4} q m^{2}$ and width $6 p^{2} q$ m. Find its length.
Q3Exponentiation in Algebraic Expressions

Simplify, leaving no brackets and no negative exponents.
1. $(3 x^{4})^{3}$
2. $\frac{(2 a^{3} b)^{2}}{4 a^{2} b}$
3. ${(\frac{m^{5}}{m^{2}})}^{4}$
4. A storage cube has an edge of $4 t^{3}$ cm. Write its volume.
Q4Factoring Out a Greatest Common Factor

Factor each expression by taking out the greatest common factor.
1. $12 x^{3} + 18 x^{2}$
2. $20 a^{2} b - 35 a b^{2} + 15 a b$
3. A rectangular banner has an area of $(24 n^{2} + 40 n) {cm}^{2}$ , and its width in centimetres is the greatest common factor of the two terms. Find the width and the length.
Q5Factoring Polynomials

Answer each part.
1. Explain what it means to say a polynomial is written in factored form, and say why $3 x + 12 x^{2}$ is not in factored form.
2. For each factoring below, name the technique used — taking out a common factor, or grouping — and say whether the factoring is complete.
Q6Factoring a Monomial

Answer each part.
1. Write $36 x^{5} y^{2}$ as a product of two monomials, one of which is $9 x^{2} y$ .
2. Find the missing monomial: $(- 5 a^{3}) () = 40 a^{7} b$ .
3. A cube-shaped crate has a volume of $27 k^{9} {cm}^{3}$ . Find the length of one edge.
Q7Factoring by Grouping

Factor each polynomial by grouping.
1. $x^{3} + 4 x^{2} + 3 x + 12$
2. $6 a b + 9 a + 10 b + 15$
3. $12 m n - 8 m + 15 n - 10$
Q8From Monomials to Polynomials

For each polynomial, give the number of terms, its name (monomial, binomial, trinomial or polynomial), its degree, and the coefficient of its term of highest degree. Then rewrite it in decreasing order of degree.
1. $5 - 3 x^{2} + 8 x$
2. $7 m^{3} n^{2}$
3. $4 y - y^{4}$
4. $- 6$
Q9Multiplying Algebraic Expressions

Expand and simplify.
1. $(x + 7) (x - 3)$
2. $(2 a - 5) (3 a + 4)$
3. $(3 m + 2 n) (4 m - n)$
4. A rectangular herb garden measures $(x + 6)$ m by $(x + 2)$ m. Write its area in expanded form and its perimeter in simplified form.
Q10Operations on Algebraic Expressions

Decide whether each statement is true for every value of the variable that lets both sides be worked out. If it is true, name the rule or property that makes it work. If it is false, give a value of the variable that shows it fails and write the correct right-hand side.
1. $x^{4} \cdot x^{3} = x^{12}$
2. $(2 x^{3})^{2} = 4 x^{6}$
3. $\frac{9 x^{5} + 6 x^{2}}{3 x^{2}} = 3 x^{3} + 2$
4. $5 x^{2} + 3 x^{2} = 8 x^{4}$
5. $6 x + 9 = 3 (2 x + 3)$
Q11Subtracting algebraic expressions

Answer both parts.
1. Simplify $(9 x^{3} - 2 x + 7) - (4 x^{3} + 5 x^{2} - 2 x - 1)$ .
2. Ordering $j$ jerseys costs Team Nord $32 j + 250$ dollars and Team Sud $27 j + 400$ dollars. Write a simplified expression for how much more Team Nord pays, then evaluate it for $j = 40$ .
Q12Synthesis — drawing on several sheets in this topic

A rectangular loading platform has an area of $(18 x^{3} + 30 x^{2}) m^{2}$ and a width of $6 x^{2}$ m.
1. Find its length by dividing.
2. Factor $18 x^{3} + 30 x^{2}$ completely, and say how the factoring confirms your answer to part (a).
3. The platform is rebuilt $2$ m wider and $4$ m longer. Write the new area in expanded form.
4. Evaluate the new area for $x = 2$ , and check your value from the actual dimensions.

The 10 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

Which stream is this for? Secondary 3 is the common year. Everybody takes the same mathematics before the CST, TS and SN options separate in Secondary 4, so this sheet is not written for one of them — it is the ground all three stand on. That also makes it a fair signal: if the factoring here feels shaky, that is worth knowing while you are still choosing.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

What this year actually adds

Half of this topic is Secondary 2 material carried forward — naming a monomial, adding and subtracting polynomials, substituting a value. The other half is new, and the new half is the part that gets used constantly from here on. It is worth knowing which is which.

The four new moves, in the order the sheet meets them

Each one is a way of taking an expression apart rather than tidying it up.

1
Dividing a polynomial by a monomial
Every term of the numerator gets divided, not just the first. Q2 is built so that one term divides into itself and leaves 1 behind — the single most common place a term disappears from an answer.
2
Raising a product to a power
Q3 applies the exponent to the coefficient as well as to the letters. (3x⁴)³ is 27x¹², not 3x¹², and the coefficient is where most of the marks are lost.
3
Factoring out a greatest common factor
Q4, with Q6 running the same arithmetic on a single monomial — splitting one term into a product, and recovering a missing factor. The word doing the work is greatest: an answer whose multiplication checks out perfectly can still be unfinished, and the challenge set makes exactly that point.
4
Factoring by grouping, and multiplying two binomials
Q7 and Q9, and they are the same fact read in opposite directions. Multiplying (x + 7)(x − 3) builds four products; grouping starts from four terms and rebuilds the two brackets they came from.

Factoring is multiplication read backwards, so check it that way

There is no such thing as an unverifiable factoring. You produced a product; multiply it out and see whether the original comes back. Doing that turns a whole class of errors into a ten-second check, and Q5 asks you to make the judgement explicitly — to name the technique used and say whether the factoring is complete.

Two different ways a factoring goes wrong, and only one of them is caught by expanding.

A term went missing. Writing 16x⁴ − 24x³ + 8x² = 8x²(2x² − 3x) loses the last term, because 8x² divided by itself is 1 and the 1 was never written down. Expanding catches this immediately: three terms went in and two came out.

The factor taken out was not the greatest one. 30m³ − 45m² + 60m = 5m(6m² − 9m + 12) multiplies back perfectly and is still not finished, because the bracket has a common factor of 3 left in it. Expanding will never tell you this. The test is to look inside the bracket afterwards and ask whether its terms still share anything — a number, a letter, or both.

Grouping: pair them, and the two pairs must agree

Four terms, taken two at a time. Factor each pair on its own, and if the same bracket falls out of both, that bracket comes out and what is left of the two pairs forms the second one.

x³ + 4x² + 3x + 12 = x²(x + 4) + 3(x + 4) = (x + 4)(x² + 3)

Three things decide whether it works, and each of them is a separate question in the challenge set:

The sign belongs to the factor you take out. With 10xy − 4x − 15y + 6, the second pair needs −3 taken out, not 3 — pulling out the negative is what flips −15y + 6 into the same bracket the first pair produced. Take out a positive and the two brackets disagree, and it looks as though the polynomial does not factor at all.

The order of the terms is yours to choose. 2a³ + 15 + 10a² + 3a refuses to group as written, and the same four terms in a different order group without trouble. If a pairing fails, rearrange before you conclude anything.

Some polynomials genuinely do not group. That is a legitimate answer, but it is only an answer once you have tried the pairings and shown what goes wrong in each. "I could not do it" and "it cannot be done" are different claims, and the second one has to be earned.

An exponent spreads over a product. It does not spread over a sum.

This is the most expensive misconception in the topic, and the challenge question on exponentiation is built around it. From (2x)³ = 8x³ and (3x)² = 9x² a student concludes that the power simply visits each piece — and then writes (x + 4)² = x² + 16, which is false for every value of x but one.

Why the two cases differ. A power is repeated multiplication, so (2x)³ is 2x · 2x · 2x, and multiplication lets you reshuffle the factors into (2 · 2 · 2)(x · x · x). There is nothing to reshuffle in (x + 4)(x + 4) — it is a multiplication of two sums, so every term of the first meets every term of the second, and the two cross products are the 8x that the wrong answer throws away.

One counterexample settles it. At x = 1, (1 + 4)² = 25 while 1² + 16 = 17. That is a complete disproof, and it is the form the question wants.

Multiplying two binomials, and the frame problem

Q9 is four products and then a collection of like terms — (2a − 5)(3a + 4), (3m + 2n)(4m − n), and a garden whose area and perimeter both have to be written from the same two dimensions. Notice that the area needs a multiplication and the perimeter needs an addition; students routinely expand both.

The same two-dimensions-one-variable shape returns whenever a uniform border runs round a rectangle, and the trap is the same every time it appears — a border of 3 cm runs down both sides, so each dimension grows by 6, not by 3. Draw the rectangle, mark the border on all four edges, and the doubling stops being something you have to remember.

Degree, names, and the true-or-false question

Q8 asks for the number of terms, the name, the degree and the leading coefficient of four polynomials, one of which is the bare constant −6. A constant is a monomial of degree zero, and that is worth saying out loud once, because a zero-degree function comes back as a whole sheet in the functions topic.

Q10 gives five statements and asks whether each holds for every value of the variable. When a statement is true, name the property; when it is false, produce a number that breaks it and write the correct right-hand side. Both halves are the answer. x⁴ · x³ = x¹² fails because exponents add when powers of the same base are multiplied, and 5x² + 3x² = 8x⁴ fails because adding like terms adds the coefficients and leaves the exponent alone. Those two errors are opposites, and meeting them side by side is the point of the question.

The synthesis question

A loading platform whose area is 18x³ + 30x² and whose width is 6x². You divide to get the length, then factor the same polynomial completely and notice that the factoring contains the division you just did. The platform is then rebuilt wider and longer, which turns both dimensions into binomials and sends you back to multiplying them. It is the whole topic in one question, and the last part asks you to check your expanded area against the actual measurements for a particular value — a habit worth keeping.

Getting the most out of it

Expand every factoring you write

Every single one, at least until it stops catching anything. It costs a line, and it converts guessing into checking. Then look inside the bracket you produced and ask whether its terms still share a factor — that second look is what separates a correct factoring from a complete one.

Draw the rectangle

Whenever a question mentions an area, a width, a border or a frame, sketch it and write the dimensions on the edges. The border questions in particular are almost impossible to get wrong from a picture and easy to get wrong from a sentence.

Do the true-or-false question twice

Answer it cold before anything else and mark which statements you were unsure about. Answer it again at the end of the sheet. The ones you changed your mind about are the rules that had not settled yet.

Treat this sheet as information about next year

Secondary 3 is the common year, and factoring is what Secondary 4 leans on hardest in every option. If this material takes real effort now, that is not a verdict — it is a useful, early piece of evidence to have while the option choice is still open.

Want the solutions, or something more challenging?

The worksheet, the questions above and every explanation on this page stay free permanently. Three more PDFs exist for this topic — the worked answer key, a harder problem set, and the answer key to that. They come with the Secondary 3 Math Solutions Bundle, which is what keeps the rest of the series free.

What else exists for Algebraic Expressions

Three PDFs · 16 pages · all three are in the bundle below.

Answer key — 3 pages. All 12 questions worked step by step, including the restrictions and the justifications. Not a list of final answers.
Challenge problems — 10 pages, 10 problems. A separate sheet at exam-plus difficulty covering the same 11 concepts. Harder than anything on the free sheet.
Challenge answer key — 3 pages. Every challenge problem worked to the same standard, with the checks shown.
PDF, letter size, print-ready.

In the bundle See what's in it Not sold separately

The one thing that's for sale

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Every Secondary 3 Math topic — the complete Solutions Bundle

One download, one payment, the whole program. Every answer key and every challenge set for all 11 Secondary 3 Math worksheet sets — including this one.

11 sets · 33 PDFs · 154 pages$19.99

Worked solutions, not answer lists — every step written out
Covers the whole year's program at this level
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Taking Secondary 1 Math as well? The Secondary 1 Math bundle covers all 15 of its sets — 45 PDFs, 182 pages — on the same terms.

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Common questions

Is this worksheet really free?

Yes — the questions are on this page to read, and the PDF downloads directly, no email and no account. The one paid item is optional: the complete Secondary 3 Solutions Bundle, which covers every set at this level.

What do I get if I buy the bundle?

Every answer key and every challenge set for the whole Secondary 3 series, in one download. For this topic that is the worked answer key to the practice questions, a set of harder problems, and the worked answer key to those. The practice worksheets themselves stay free.

How do I know when a factoring is finished?

Multiply your answer out and check that the original comes back — that catches a lost term. Then look inside the bracket and ask whether its terms still share a number or a letter. A factoring can multiply back perfectly and still be unfinished, because the factor taken out was not the greatest one.

Why does factoring by grouping sometimes not work?

Usually because of the pairing or the sign. The two pairs have to leave the same bracket behind, and getting there often means taking a negative out of the second pair, or reordering the four terms first. Some polynomials really do not group, but that is only a conclusion once you have tried more than one pairing and can say what goes wrong each time.

Is it true that (x + 4) squared equals x squared plus 16?

No, and one number shows it: at x = 1 the left side is 25 and the right side is 17. An exponent spreads over a product because a power is repeated multiplication and the factors can be regrouped. A sum has nothing to regroup, so the two binomials have to be multiplied out term by term, and the cross terms the shortcut discards are exactly what is missing.

Which Secondary 3 stream is this for?

There is no stream yet. Secondary 3 is the common year that every student takes before the CST, TS and SN options separate in Secondary 4, so this sheet is written for all of them. Factoring and multiplying binomials are assumed by all three, which is why the topic is worth the effort now.

Can teachers use this in class?

Yes. Print and photocopy it for your own classes freely — I just ask that the tutorinmontreal.ca footer stays on the page.

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Yes — through one-on-one tutoring, in Montreal or online. Get in touch to arrange a session, or see the current rates.

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The same topic at the other level: Secondary 4 Math · Algebraic Expressions.