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Year 9 · Mathematics · Algebra

Expanding binomial products

Expand products of two binomials such as (x + 3)(x + 5), and recognise the special patterns for perfect squares and the difference of two squares.

Awaiting educator review. This material was drafted by our content team with AI assistance and is waiting for review by a qualified educator. Please check it suits your child before using it. Created 10 October 2026.

Learning goals

Learners will

  • Expand the product of two binomials using the distributive law.
  • Collect like terms to simplify an expansion.
  • Recognise perfect squares and the difference of two squares.

Success looks like

  • I can expand (x + a)(x + b) and collect like terms.
  • I can expand a binomial with a coefficient, such as (2x − 1)(x + 4).
  • I can expand (x + a)^2 and (x − a)(x + a) and describe the patterns.

The big idea

A binomial has two terms, such as x + 3. To multiply two binomials, every term in the first bracket must multiply every term in the second bracket. That gives four products, which are then simplified by collecting like terms.

One helpful picture is an area diagram. Draw a rectangle with sides x + 3 and x + 5, split into four smaller rectangles. Their areas are x^2, 5x, 3x and 15. Adding them gives x^2 + 8x + 15, so (x + 3)(x + 5) = x^2 + 8x + 15.

Some learners remember the four products using FOIL: First, Outer, Inner, Last. In (x + 3)(x + 5) these are x × x, x × 5, 3 × x and 3 × 5. FOIL is only a memory aid; the underlying idea is the distributive law.

Take care with negative signs. Treat each sign as belonging to the term after it. In (x − 3)(x + 7), the terms are x and −3 in the first bracket, so the products are x^2, 7x, −3x and −21, giving x^2 + 4x − 21.

Perfect squares: (x + a)^2 means (x + a)(x + a), which always expands to x^2 + 2ax + a^2. The middle term is double the product of the two terms. A very common mistake is to write (x + 5)^2 = x^2 + 25, forgetting the middle term 10x.

Difference of two squares: (x − a)(x + a) expands to x^2 − a^2, because the two middle terms (−ax and +ax) cancel out. For example, (x − 4)(x + 4) = x^2 − 16.

Worked examples

Two simple binomials

Expand and simplify (x + 3)(x + 5).

  1. Multiply each term in the first bracket by each term in the second: x × x = x^2, x × 5 = 5x, 3 × x = 3x, 3 × 5 = 15.
  2. Write them together: x^2 + 5x + 3x + 15.
  3. Collect like terms: 5x + 3x = 8x.

Answer: x^2 + 8x + 15

A binomial with a coefficient and a negative

Expand and simplify (2x − 1)(x + 4).

  1. Products: 2x × x = 2x^2, 2x × 4 = 8x, −1 × x = −x, −1 × 4 = −4.
  2. Write them together: 2x^2 + 8x − x − 4.
  3. Collect like terms: 8x − x = 7x.

Answer: 2x^2 + 7x − 4

A perfect square

Expand (x + 6)^2.

  1. Write it as a product: (x + 6)(x + 6).
  2. Products: x^2, 6x, 6x and 36.
  3. Collect like terms: 6x + 6x = 12x.

Answer: x^2 + 12x + 36

Practice check

Have a go, then check your answers. Each answer comes with an explanation.

  1. 1.Expand and simplify (x + 2)(x + 4).
  2. 2.Expand and simplify (x − 3)(x + 7).
  3. 3.Expand (x + 5)^2.
  4. 4.Expand (x − 4)(x + 4).
  5. 5.Expand and simplify (3x + 2)(x − 1).
  6. 6.When (x + 9)(x − 2) is expanded and simplified, what is the coefficient of x?
  7. 7.When (x − 6)(x − 8) is expanded, what is the constant term (the term with no x)?
  8. 8.When (2x + 5)^2 is expanded and simplified, what is the coefficient of x?
Answer guide for parents

  1. Expand and simplify (x + 2)(x + 4).

    x^2 + 6x + 8 — The products are x^2, 4x, 2x and 8. Collecting like terms gives x^2 + 6x + 8.

  2. Expand and simplify (x − 3)(x + 7).

    x^2 + 4x − 21 — The products are x^2, 7x, −3x and −21. Since 7x − 3x = 4x, the answer is x^2 + 4x − 21.

  3. Expand (x + 5)^2.

    x^2 + 10x + 25 — (x + 5)(x + 5) gives x^2 + 5x + 5x + 25 = x^2 + 10x + 25.

  4. Expand (x − 4)(x + 4).

    x^2 − 16 — This is a difference of two squares. The middle terms −4x and +4x cancel, leaving x^2 − 16.

  5. Expand and simplify (3x + 2)(x − 1).

    3x^2 − x − 2 — The products are 3x^2, −3x, 2x and −2. Since −3x + 2x = −x, the answer is 3x^2 − x − 2.

  6. When (x + 9)(x − 2) is expanded and simplified, what is the coefficient of x?

    7 — The x terms are −2x and 9x, which add to 7x. So the coefficient is 7.

  7. When (x − 6)(x − 8) is expanded, what is the constant term (the term with no x)?

    48 — The constant term is (−6) × (−8) = 48. A negative times a negative is positive.

  8. When (2x + 5)^2 is expanded and simplified, what is the coefficient of x?

    20 — (2x + 5)(2x + 5) = 4x^2 + 10x + 10x + 25 = 4x^2 + 20x + 25, so the coefficient of x is 20.

Watch out for

  • Only multiplying the first terms and the last terms, such as (x + 5)^2 = x^2 + 25. An area diagram shows the two missing rectangles clearly.
  • Losing negative signs. Encourage circling each term together with the sign in front of it before multiplying.
  • Trying to collect unlike terms, such as adding x^2 and x. Remind learners that only terms with exactly the same pronumeral part can be combined.

Tips for parents

  • Ask your learner to draw an area diagram for the first few questions. It makes the four products visible and reduces errors.
  • Check answers by substituting a number. For example, with x = 1, (x + 3)(x + 5) = 4 × 6 = 24 and x^2 + 8x + 15 = 1 + 8 + 15 = 24.
  • Praise neat, step-by-step setting out. Most errors in this topic come from rushing.

Go further

  • Use the difference of two squares to calculate 31 × 29 mentally, by writing it as (30 + 1)(30 − 1) = 900 − 1 = 899. Create your own examples.
  • Cut a square of paper with side length x + 3 into pieces that show x^2, two 3x rectangles and a 3 × 3 square.
  • Expand (x + 1)^3 by first expanding (x + 1)^2 and then multiplying by (x + 1). Look for a pattern in the coefficients.