Algebraic Expressions

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Quick recall

• Expand brackets carefully, especially when subtracting a bracket.
• Only terms with the same variables and powers can be combined.

On this page

Formula Cards

1. Combine like terms

$$3x+5x=8x$$

Combine only terms with matching variables and powers.

2. Square of a sum

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

3. Square of a difference

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

4. Difference of squares

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

5. Three-term square

$$(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca$$

6. Sum of cubes

$$a^3+b^3=(a+b)(a^2-ab+b^2)$$

7. Difference of cubes

$$a^3-b^3=(a-b)(a^2+ab+b^2)$$

8. Factorization reverses expansion

$$x^2-9=(x-3)(x+3)$$

Worked Examples

Example 1

Question: Simplify 3(x+2)−2(x−1).

Step 1: Expand both brackets.

$$3x+6-2x+2$$

Step 2: Collect like terms.

$$(3x-2x)+(6+2)=x+8$$

Answer: x+8.

Example 2

Question: Expand (x+5)².

Step 1: Use the square-of-a-sum identity.

$$(x+5)^2=x^2+2(x)(5)+5^2$$

Step 2: Simplify.

$$x^2+10x+25$$

Answer: x²+10x+25.

Example 3

Question: Factor x²−16.

Step 1: Recognize a difference of squares.

$$x^2-16=x^2-4^2$$

Step 2: Use the identity.

$$x^2-4^2=(x-4)(x+4)$$

Answer: (x−4)(x+4).

Example 4

Question: Simplify (2x+3)(2x−3).

Step 1: Recognize a difference of squares.

$$(2x+3)(2x-3)=(2x)^2-3^2$$

Step 2: Evaluate the squares.

$$4x^2-9$$

Answer: 4x²−9.

Drill

Try each question before opening its answer.

Question 1

Expand $(a-4)^2$.

• Show answer

  a²-8a+16

Question 2

Factor $x^2-25$.

• Show answer

  (x-5)(x+5)

Question 3

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

• Show answer

  5x

Question 4

Expand $(x+2)(x+5)$.

• Show answer

  x²+7x+10

Question 5

Factor $x^2+7x+12$.

• Show answer

  (x+3)(x+4)

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