50-minute lesson
Year 9 Mathematics: Expanding binomial products
Lesson objective
- 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 criteria:
- 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.
Materials
- Paper and pencil
- Grid paper or a ruler for drawing area diagrams
- Scissors (optional)
Introduction
5 minIntroduce today's words:
- Binomial: An algebraic expression with two terms, such as x + 3 or 2x − 1.
- Expand: Remove the brackets by multiplying out, writing the expression as a sum of terms.
- Distributive law: The rule a(b + c) = ab + ac: the term outside multiplies every term inside.
- Like terms: Terms with exactly the same pronumeral part, such as 3x and −5x, which can be added or subtracted.
- Difference of two squares: The pattern (a − b)(a + b) = a^2 − b^2.
Ask your child what they already know about expanding binomial products.
Explanation
10 minA 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
8 minTwo simple binomials
Expand and simplify (x + 3)(x + 5).
Step 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.
Step 2: Write them together: x^2 + 5x + 3x + 15.
Step 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).
Step 1: Products: 2x × x = 2x^2, 2x × 4 = 8x, −1 × x = −x, −1 × 4 = −4.
Step 2: Write them together: 2x^2 + 8x − x − 4.
Step 3: Collect like terms: 8x − x = 7x.
Answer: 2x^2 + 7x − 4
A perfect square
Expand (x + 6)^2.
Step 1: Write it as a product: (x + 6)(x + 6).
Step 2: Products: x^2, 6x, 6x and 36.
Step 3: Collect like terms: 6x + 6x = 12x.
Answer: x^2 + 12x + 36
Guided practice (do together)
10 min1. Expand and simplify (x + 2)(x + 4).
(a) x^2 + 8x + 6 (b) x^2 + 6x + 8 (c) x^2 + 8 (d) x^2 + 6x + 6
2. Expand and simplify (x − 3)(x + 7).
(a) x^2 − 4x − 21 (b) x^2 + 4x + 21 (c) x^2 + 4x − 21 (d) x^2 − 21
3. Expand (x + 5)^2.
(a) x^2 + 10x + 25 (b) x^2 + 25 (c) x^2 + 5x + 25 (d) 2x + 10
4. Expand (x − 4)(x + 4).
(a) x^2 − 8x − 16 (b) x^2 + 16 (c) x^2 − 8 (d) x^2 − 16
Independent practice
12 min5. Expand and simplify (3x + 2)(x − 1).
(a) 3x^2 + x − 2 (b) 3x^2 − x − 2 (c) 3x^2 − 5x − 2 (d) 4x^2 − x − 2
6. When (x + 9)(x − 2) is expanded and simplified, what is the coefficient of x?
7. When (x − 6)(x − 8) is expanded, what is the constant term (the term with no x)?
8. When (2x + 5)^2 is expanded and simplified, what is the coefficient of x?
Questions to check understanding
- Can you expand (x + a)(x + b) and collect like terms?
- Can you expand a binomial with a coefficient, such as (2x − 1)(x + 4)?
- Can you expand (x + a)^2 and (x − a)(x + a) and describe the patterns?
- What was the trickiest part today?
Answer guide
1. x^2 + 6x + 8 — The products are x^2, 4x, 2x and 8. Collecting like terms gives x^2 + 6x + 8.
2. 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. x^2 + 10x + 25 — (x + 5)(x + 5) gives x^2 + 5x + 5x + 25 = x^2 + 10x + 25.
4. x^2 − 16 — This is a difference of two squares. The middle terms −4x and +4x cancel, leaving x^2 − 16.
5. 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. 7 — The x terms are −2x and 9x, which add to 7x. So the coefficient is 7.
7. 48 — The constant term is (−6) × (−8) = 48. A negative times a negative is positive.
8. 20 — (2x + 5)(2x + 5) = 4x^2 + 10x + 10x + 25 = 4x^2 + 20x + 25, so the coefficient of x is 20.
Review
5 minRecap the success criteria together. Watch for these common misconceptions:
- 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.
Extension activities
- 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.
Suggested follow-up
- Revisit expanding binomial products tomorrow with two or three quick questions from memory.
- Try the online practice check for this topic and look at any questions that need another go.
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.