Simplifying Algebraic Expressions

Simplifying an expression means rewriting it in a shorter, tidier form that is still equivalent: it gives the same value for every value of the variable. A simpler expression is easier to read, easier to evaluate, and much easier to use when you solve equations. The two big tools are collecting like terms and the distributive property.

Key ideas

Like terms

Like terms have exactly the same variable part: the same letters, raised to the same exponents. Only the coefficients can be different.

Like terms	Not like terms
$3x$ and $-5x$	$3x$ and $3x^2$ (different exponents)
$4a^2$ and $a^2$	$2a$ and $2b$ (different letters)
$7$ and $-2$ (both constants)	$5x$ and $5$ (one has no variable)
$2xy$ and $-yx$ (order doesn’t matter)	$xy$ and $x$

Collecting like terms

To collect (combine) like terms, add their coefficients and keep the variable part the same:

$$5x + 3x = 8x \qquad 4y - 9y = -5y \qquad x + x = 2x$$

Think of $x$ as a type of object. Five apples plus three apples is eight apples, but five apples plus three bananas is just “five apples plus three bananas”. In the same way, $3x + 2$ can’t be simplified: $3x$ and $2$ are not like terms.

Algebra tiles

Algebra tiles make this concrete. A long tile stands for $x$, a small square tile stands for $1$, and a large square tile stands for $x^2$. Tiles of a different colour (often red) stand for negatives, and a positive and a negative tile of the same kind make a zero pair that cancels out.

Collecting like terms is just sorting the tiles by type and counting each type.

The distributive property

To multiply a number by a bracket, multiply it by every term inside:

$$a(b + c) = ab + ac$$

The algebra tiles show why. $2(x + 3)$ means two groups of $x + 3$. Put the groups together, and you have 2 $x$-tiles and 6 unit tiles:

Two groups of $x + 3$ regroup into $2x + 6$, so $2(x + 3) = 2x + 6$.

Distributing a negative

Be extra careful when the number in front is negative. The negative multiplies every term, so every sign inside changes:

$$-3(x - 4) = -3x + 12 \qquad -(x + 5) = -x - 5$$

A minus sign in front of a bracket, like $-(x + 5)$, means $-1(x + 5)$.

Simplifying expressions with brackets

1. Expand the brackets using the distributive property.
2. Collect like terms.

It’s a good habit to check your answer by substituting a number into the original expression and into your answer. They should match.

Multiplying and dividing monomials

A monomial is a single term, like $3x^2$ or $-5y$. To multiply or divide monomials, deal with the numbers and the variables separately, and use the exponent laws for the variables:

$$(3x^2)(4x^3) = (3 \times 4)(x^2 \times x^3) = 12x^5$$ $$\frac{12a^5}{4a^2} = \frac{12}{4} \times \frac{a^5}{a^2} = 3a^3$$

When multiplying, add the exponents. When dividing, subtract them.

To multiply a monomial by a binomial, use the distributive property together with the exponent laws:

$$2x(3x - 5) = 6x^2 - 10x$$

In Grade 10 you’ll go further and multiply two binomials, in polynomial operations.

Worked examples

Example 1: Collecting like terms

Simplify.

• (a) $5x + 3 - 2x + 7$
• (b) $4a^2 - 3a + a^2 + 8a - 1$

Solution.

(a) Group the $x$ terms and the constants (keep each sign with its term):

$$\begin{aligned} 5x + 3 - 2x + 7 &= 5x - 2x + 3 + 7 \\ &= 3x + 10 \end{aligned}$$

(b) Group the $a^2$ terms, the $a$ terms, and the constants:

$$\begin{aligned} 4a^2 - 3a + a^2 + 8a - 1 &= 4a^2 + a^2 - 3a + 8a - 1 \\ &= 5a^2 + 5a - 1 \end{aligned}$$

$5a^2$ and $5a$ are not like terms, so this is as simple as it gets.

Example 2: Distributing, including negatives

Expand.

• (a) $4(2x - 3)$
• (b) $-2(5 - 3y)$
• (c) $-(4m - 1)$

Solution.

(a) Multiply each term by $4$:

$$4(2x - 3) = 4(2x) + 4(-3) = 8x - 12$$

(b) Multiply each term by $-2$. A negative times a negative is positive:

$$-2(5 - 3y) = (-2)(5) + (-2)(-3y) = -10 + 6y$$

You can also write this as $6y - 10$.

(c) The minus sign means $-1$, so every sign changes:

$$-(4m - 1) = -4m + 1$$

Example 3: Brackets, then like terms

Simplify $3(2x + 1) - 2(x - 4)$.

Solution. Expand each bracket. The second bracket is multiplied by $-2$, not $2$:

$$\begin{aligned} 3(2x + 1) - 2(x - 4) &= 6x + 3 - 2x + 8 \\ &= 4x + 11 \end{aligned}$$

Check with $x = 1$. Original: $3(3) - 2(-3) = 9 + 6 = 15$. Answer: $4(1) + 11 = 15$. ✓

Example 4: Monomials

Simplify.

• (a) $(-2x^3)(5x^4)$
• (b) $\dfrac{18y^6}{-3y^2}$
• (c) A rectangle is $3x$ cm wide and $(2x + 5)$ cm long. Write a simplified expression for its area.

Solution.

(a) Multiply the numbers, then add the exponents:

$$(-2x^3)(5x^4) = (-2 \times 5)(x^{3 + 4}) = -10x^7$$

(b) Divide the numbers, then subtract the exponents:

$$\frac{18y^6}{-3y^2} = \frac{18}{-3} \times y^{6 - 2} = -6y^4$$

(c) Area is length times width:

$$3x(2x + 5) = 3x(2x) + 3x(5) = 6x^2 + 15x$$

The area is $6x^2 + 15x$ square centimetres. Check with $x = 2$: the rectangle is $6$ cm by $9$ cm, with area $54$ cm², and $6(4) + 15(2) = 24 + 30 = 54$. ✓

Common mistakes

Combining terms that aren’t alike. $3x + 2$ is not $5x$, and $4x + 3x^2$ is not $7x^3$. Only terms with exactly the same variable part combine. If you’re unsure, substitute a number: they happen to match at $x = 1$ (both are $5$), but at $x = 2$, $3x + 2 = 8$ while $5x = 10$.

Distributing to only the first term. $4(x + 3)$ is $4x + 12$, not $4x + 3$. The number outside multiplies everything inside the bracket.

Losing a sign when distributing a negative. In $5 - 2(x - 4)$, the $-2$ multiplies both terms: $-2x + 8$. A very common error is writing $-2x - 8$. Remember that a negative times a negative is positive.

Multiplying exponents instead of adding them. $x^2 \times x^3 = x^5$, not $x^6$. Write it out if you need to: $(x \cdot x)(x \cdot x \cdot x)$ has five factors of $x$.

Thinking x + x is x². Adding gives $x + x = 2x$ (two $x$‘s). Multiplying gives $x \cdot x = x^2$. Also remember that $x$ by itself has coefficient $1$, so $x + 4x = 5x$.

Practice

1. (Warm-up) Which of these are like terms with $4x$?

$$7x \qquad 4x^2 \qquad -x \qquad 4 \qquad \frac{x}{2}$$

Solution

$7x$, $-x$ and $\dfrac{x}{2}$ are like terms with $4x$ (note that $\dfrac{x}{2}$ is the same as $\dfrac{1}{2}x$). $4x^2$ has a different exponent, and $4$ has no variable.

2. (Warm-up) Simplify.

• (a) $6m - 2m + m$
• (b) $3x + 4 - x - 9$
• (c) $2a + 3b - 5a + b$

Solution

(a) $6m - 2m + m = (6 - 2 + 1)m = 5m$

(b) $3x - x + 4 - 9 = 2x - 5$

(c) $2a - 5a + 3b + b = -3a + 4b$

3. (Warm-up) Expand.

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

Solution

(a) $5x + 10$

(b) $(-3)(2y) + (-3)(-1) = -6y + 3$

(c) $-a + 7$

4. (Core) Simplify $4(x - 2) + 3(2x + 5)$.

Solution

$$\begin{aligned} 4(x - 2) + 3(2x + 5) &= 4x - 8 + 6x + 15 \\ &= 10x + 7 \end{aligned}$$

Check with $x = 1$: $4(-1) + 3(7) = -4 + 21 = 17$, and $10 + 7 = 17$. ✓

5. (Core) Simplify $2(3k - 1) - 5(k - 2)$.

Solution

The second bracket is multiplied by $-5$:

$$\begin{aligned} 2(3k - 1) - 5(k - 2) &= 6k - 2 - 5k + 10 \\ &= k + 8 \end{aligned}$$

Check with $k = 2$: $2(5) - 5(0) = 10$, and $2 + 8 = 10$. ✓

6. (Core) Simplify.

• (a) $(4p^2)(-3p^5)$
• (b) $\dfrac{-20x^8}{5x^3}$
• (c) $2x(4x - 3)$

Solution

(a) $(4)(-3)\,p^{2 + 5} = -12p^7$

(b) $\dfrac{-20}{5}\,x^{8 - 3} = -4x^5$

(c) $2x(4x) + 2x(-3) = 8x^2 - 6x$

7. (Core) A triangle has sides of length $(x + 4)$ cm, $(2x - 1)$ cm and $3x$ cm.

• (a) Write a simplified expression for its perimeter.
• (b) Find the perimeter when $x = 5$.

Solution

(a) Add the three sides and collect like terms:

$$(x + 4) + (2x - 1) + 3x = x + 2x + 3x + 4 - 1 = 6x + 3$$

The perimeter is $(6x + 3)$ cm.

(b) $6(5) + 3 = 33$, so the perimeter is $33$ cm.

Check: the sides are $9$, $9$ and $15$ cm, and $9 + 9 + 15 = 33$. ✓

8. (Challenge) Simplify $x(x + 3) - 2(x^2 - x)$.

Solution

$$\begin{aligned} x(x + 3) - 2(x^2 - x) &= x^2 + 3x - 2x^2 + 2x \\ &= x^2 - 2x^2 + 3x + 2x \\ &= -x^2 + 5x \end{aligned}$$

Check with $x = 2$: $2(5) - 2(4 - 2) = 10 - 4 = 6$, and $-4 + 10 = 6$. ✓

9. (Challenge) Find the numbers $a$ and $b$ that make this true for every value of $x$:

$$3(2x - a) + bx = 10x - 12$$

Solution

Expand the left side and collect like terms:

$$3(2x - a) + bx = 6x - 3a + bx = (6 + b)x - 3a$$

For this to equal $10x - 12$ for every $x$, the $x$ terms must match and the constants must match:

• $6 + b = 10$, so $b = 4$.
• $-3a = -12$, so $a = 4$.

Check: $3(2x - 4) + 4x = 6x - 12 + 4x = 10x - 12$. ✓