Secondary 2 Algebraic Expressions Worksheet

All 10 questions

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

1. Q1Adding Algebraic Expressions

   A community centre in Trois-Rivières rents out two rooms. Renting the dance studio for $h$ hours costs $18h+40$ dollars, and renting the games room for $h$ hours costs $25h+15$ dollars.

   1. Write a simplified expression for the cost of renting both rooms for $h$ hours.
   2. Use your expression to find the cost of renting both rooms for $3$ hours.
2. Q2Algebra - Algebraic Expressions

   At a bubble-tea shop a plain cup costs $c$ dollars and each topping costs $t$ dollars.

   1. Write an algebraic expression, in expanded form, for the cost of $3$ cups that each have $2$ toppings.
   2. For the expression $7x^2-4x+9$, state the number of terms, the coefficient of $x$, the constant term and the degree.
3. Q3Dividing an Algebraic Expression by a Monomial

   Simplify each quotient.

   1. ${\displaystyle \frac{24x^5}{6x^2}}$
   2. ${\displaystyle \frac{15a^{3}b-20a^2{b}^2}{5a^{2}b}}$
   3. A rectangular banner has an area of $36p^{3}q{\text{cm}}^2$ and a width of $9pq$ cm. Find its length.
4. Q4Factoring a Monomial

   Answer each part.

   1. Write $18x^{3}y$ as a product of two monomials, one of which is $6xy$.
   2. A rectangle has an area of $30a^{2}b{\text{cm}}^2$ and one side measuring $5ab$ cm. Find the other side.
   3. Write $24m^4$ as a product of three monomial factors, none of which is $1$.
5. Q5From Monomials to Polynomials

   For each expression below, give the number of terms, its name (monomial, binomial, trinomial, or polynomial), and its degree.

   1. $7x$
   2. $3a^2-5a$
   3. $4$
   4. $2m^{2}n+m-6$
   5. $-9k^3+k^2-k+1$
6. Q6Multiplying Algebraic Expressions

   Expand and simplify.

   1. $(4x^2)(-3xy)$
   2. $5a(2a^2-3a+7)$
   3. A rectangular vegetable plot is $3n$ metres wide and $(2n+5)$ metres long. Write its area in expanded form.
7. Q7Operations on Algebraic Expressions

   For each statement, say whether it is true for every value of the variables. If it is true, name the property or rule that makes it work; if it is false, give a value that shows it fails and write the correct right-hand side.

   1. $3x+4y=7xy$
   2. $5a^2\times 2a^3=10a^5$
   3. $8m-3m=5m$
   4. ${\displaystyle \frac{6x+9}{3}}=2x+9$
   5. $2(x+5)=2x+10$
8. Q8Simplifying Algebraic Expressions

   Simplify each expression by collecting like terms.

   1. $9x-4+3x^2-7x+11-x^2$
   2. $6a+2(3a-5)-4$
   3. $5mn+3m-2mn+m^2$
9. Q9Subtracting algebraic expressions

   Answer both parts.

   1. Simplify $(8x^2+5x-3)-(2x^2-4x+9)$.
   2. Plan A for a phone costs $35+8g$ dollars when $g$ gigabytes of extra data are used; plan B costs $50+3g$ dollars. Write a simplified expression for how much more plan B costs than plan A, then evaluate it for $g=2$.
10. Q10Synthesis — drawing on several sheets in this topic

    A rectangular community garden plot is $4x$ metres wide and $(3x+7)$ metres long. In one corner stands a rectangular tool shed whose base has an area of $8x^2{\text{m}}^2$ and whose width is $4x$ metres.

    1. Write the area of the whole plot in expanded form.
    2. Find the length of the shed.
    3. Write a simplified expression for the area of the plot that the shed does not cover.
    4. Evaluate that area for $x=5$.

The 9 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

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.

Everything on this sheet rests on one question: are these terms alike?

A term is a coefficient and a variable part travelling together, and two terms can only be added or subtracted when their variable parts are identical — same letters, same exponents. That single test decides more marks on this topic than anything else, because the moment it is answered wrongly the rest of the work is tidy and wrong.

Collecting like terms

Only the coefficients move; the variable part is carried along untouched. 5mn and −2mn collect to 3mn, while 3m and 3mn do not collect at all — different variable parts, so they are simply two separate terms in the finished answer.

Subtraction: the minus sign belongs to every term in the bracket

Subtracting an expression means adding the opposite of all of it. Change every sign inside the second bracket, then collect:

The term that catches people is −(−9b), which becomes +9b. Distribute the subtraction on its own line before you collect anything — one extra line, and the single most common error on this topic disappears. One question on the sheet shows exactly this slip in a student's handwriting and asks you to name the rule that was broken, which is harder, and more useful, than getting the answer yourself.

Multiplying and dividing by a monomial: the word is every

Multiplying a monomial through a bracket, and dividing a polynomial by a monomial, are the same instruction read in two directions: the outside factor meets every term inside, not just the first one.

Factoring a monomial is division wearing a different hat

Asked to write 18x³y as 6xy times something, do not hunt: divide. 18 ÷ 6 = 3 and x³y ÷ xy = x², so the missing factor is 3x². The same move answers "a rectangle has this area and this side; find the other side", which is why those questions sit beside one another here.

The interesting case is when it cannot be done. A monomial can only be split off if its coefficient divides the coefficient you started with — 24x³ is not 5x times a monomial with a whole-number coefficient, because 5 is not a factor of 24. The letters almost never cause the trouble; the numbers do.

Testing values is not proving

Two of the harder questions here hand you a conjecture built on examples that all work. Examples are how you find a pattern; simplifying is how you show it. Once 4(x + 3) − 3(x + 2) has been simplified to x + 6, the claim is settled for every value at once, which no amount of substituting ever does.

The last question puts the whole topic in one plot of land

A rectangular garden with a shed in one corner: multiply to get the whole area, divide to get the missing side of the shed, subtract to get the part the shed does not cover, then substitute a number at the end. Four operations, one situation. Check the final number the slow way as well — work out the real lengths in metres and subtract the two areas — and if the two routes disagree, the algebra has a sign in it that the arithmetic does not.

Getting the most out of it

Print it and work in pen

These are short questions with a lot of sign-handling in them, and signs are exactly what gets lost when the work is done in your head. The sheet is laid out with space under each question so the distribution step and the collecting step can each have their own line.

Substitute a number to check every answer

Pick something easy — 2 usually — and put it into the original expression and into your simplified one. They must agree. This is not extra work; it is the only way to catch a dropped sign before it costs you, and several questions on the sheet are built around a student who skipped it.

Say what you are doing out loud on the bracket questions

"Minus the whole bracket, so every sign inside flips." "Times every term, not just the first." Naming the rule as you use it is what the questions asking you to diagnose someone else's work are training, and it is the fastest way to stop making the error yourself.

Common questions