KS3 · Maths

Expanding the product of two binomials

12 × 13 is 156, so why is 100 + 6 so tempting? Split it as (10 + 2)(10 + 3) and the missing products show up.

Maths · Algebra

Walk every path through (x + 10)(x − 15)

Choose a term from the first bracket, then a term from the second. Try all four paths.

Term from the first bracket → Term from the second bracket

2 × 2 = 4 partial products, and the tree ends 4 times.

On (x + 10)(x − 15). 2 branches to choose from.

Pick one term from each bracket and see what you get. Same idea as 12 × 13 = (10 + 2)(10 + 3): the four partial products are 100, 30, 20 and 6, not just 100 + 6.

Watch out: x² − 150 uses only two of the four paths. The other two give −15x and +10x, and they matter.

Maths · Algebra

Build the product in a grid

Expand (x + 8)(x + 12). Each cell is its row times its column. Fill all four, then collect the like terms.

Grid: each cell is its row times its column
×	x	12
x
8

Collect like terms: (x + 8)(x + 12) =

Type x² as x^2 if you cannot type ². Spaces do not matter.

Maths · Algebra

No grid? Split it up

Expand (4x − 8)(2x − 3). You choose the missing steps.

(4x − 8)(2x − 3) = 4x(2x − 3) − 8(2x − 3)Split the first bracket: 4x lots of the second bracket, then −8 lots of it.
missing step

Which line is step 2?

4x × 2x = 6x²4x × 2x = 8x²4x × 2x = 8x

Maths · Algebra

Squaring a binomial: find the slip

Expand (x − 5)². One line below loses the mark. Which one?

A student's working — which line goes wrong?

(x − 5)² = (x − 5)(x − 5)
x × x = x²
x × (−5) = −5x and −5 × x = −5x
−5 × −5 = −25
x² − 5x − 5x − 25
(x − 5)² = x² − 10x − 25

Maths · Algebra

Difference of two squares, or not?

Which of these products collapse neatly into a difference of two squares? Sort each one and read why.

Still to sort

Difference of two squares (0)

One term is identical in both brackets and the other makes a zero pair.

Where the line is: The middle partial products cancel, leaving a² − b².

Not a difference of two squares (0)

The middle partial products don't cancel.

Where the line is: Looking similar isn't enough: you need an identical term and a zero pair.

6 of 6 still to sort.

Maths · Algebra

Is there a shortcut?

You spot that (x + 5)(x + 4) = x² + 9x + 20: the 9 is 5 + 4 and the 20 is 5 × 4. Now you meet (2x + 4)(2x + 5).

Which idea is closest to what you would do?

Use the same pattern: add the constants for the x term, multiply them for the number, keep x². So x² + 9x + 20.Square the 2x to get 4x², then still add the constants for the x term and multiply them for the number. So 4x² + 9x + 20.Find all four partial products: 4x² + 10x + 8x + 20, which collects to 4x² + 18x + 20.

How sure are you?

I'm sureI think soI'm guessing

Use the same pattern: add the constants for the x term, multiply them for the number, keep x². So x² + 9x + 20.

This is a common misconception

That pattern was only spotted for (x + a)(x + b). Here the x terms have a 2 in front, so the answer can't start with a plain x². And there are more x partial products to count: 2x × 5 = 10x and 4 × 2x = 8x.

Square the 2x to get 4x², then still add the constants for the x term and multiply them for the number. So 4x² + 9x + 20.

This is a common misconception

Good that you took the 2 into account for x², but the x term still comes from the partial products 10x and 8x, which collect to 18x, not 9x. The add-and-multiply pattern doesn't carry over to this type of product.

Find all four partial products: 4x² + 10x + 8x + 20, which collects to 4x² + 18x + 20.

This matches the evidence

Yes. Spotting a pattern is fine, but it doesn't work for every product. The area model or the four partial products always does.

A pattern you spot in one type of product doesn't automatically work for another. When in doubt, find all four partial products.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Two terms times two terms is four products, not two.

What you need to know

A binomial is an expression with exactly two unlike terms, like 5 − 6x or x² − x.
Have a goIs 7x − 5x a binomial? Decide before you look.
No.
7x and 5x are like terms, so 7x − 5x is just 2x: one term, not two unlike terms.
Expanding means every term in one bracket multiplies every term in the other, so two binomials give four partial products.
Have a goYour classmate says 11 × 12 is 100 + 2 = 102. Write 11 and 12 as (10 + 1)(10 + 2). Which two partial products did they miss?
10 × 2 = 20 and 1 × 10 = 10, so the answer is 100 + 20 + 10 + 2 = 132.
They multiplied first-by-first and last-by-last only. Every term in one bracket has to meet every term in the other.
12 × 13 as (10 + 2)(10 + 3) has partial products 100, 30, 20 and 6, which add to 156, not just 100 + 6.
An area model puts one bracket along each side of a rectangle, so each cell is one partial product.
Add the four partial products, then collect like terms: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6.
Treat a subtraction as adding a negative: (x − 2)(x − 3) = x² − 5x + 6, because −2 × −3 = +6.
Have a goTry (x − 4)(x − 1). What is the number term in the answer, and what sign does it have?
+4, because −4 × −1 = +4. The full answer is x² − 5x + 4.
Both brackets contain a negative, and a negative times a negative is positive. The two middle products, −x and −4x, both stay negative.
To square a binomial, write it as two brackets: (x + 7)² = (x + 7)(x + 7) = x² + 14x + 49.
With numbers in front of the letters, multiply the numbers and then the letters, so 4x × 2x = 8x².
(a + b)(a − b) = a² − b², the difference of two squares, because the two middle partial products are a zero pair.
Have a goWithout a grid: what do (x + 9)(x − 9) collapse to, and which two partial products vanish?
x² − 81. The +9x and −9x cancel.
x is identical in both brackets and +9 with −9 is a zero pair, so the middle partial products cancel.
The add-and-multiply shortcut works for (x + a)(x + b), but not for products like (2x + 4)(2x + 5).

The big picture

Expanding two brackets means every term in one bracket multiplies every term in the other. Two terms times two terms gives four partial products, and then you collect the like terms.

Key points

1A binomial has exactly two unlike terms, so 5 − 6x is one and 7x − 5x is not.

2Two binomials multiply to four partial products, then you collect like terms.

3x − 2 is the same as x + (−2), so negatives follow the same method as positives.

4The brackets can be written either way round, and the terms in a bracket can be reordered: (3 − x)(x − 7) = (−x + 3)(x − 7) = −x² + 10x − 21. It is common to write the answer with the x² term first.

5The shortcut for (x + a)(x + b) doesn't work for every product. When in doubt, find all four partial products.

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Worked example

Problem

Expand (x + 5)(x − 3).

⚠ Watch out

Writing (x + 10)(x − 15) as x² − 150. That only multiplies first-by-first and last-by-last, just like 12 × 13 = 100 + 6. The other two partial products, −15x and +10x, are missing, and the answer is x² − 5x − 150.

🧠

Memory hook

Two terms times two terms is four partial products. Count the paths and nothing gets lost.

✓

Check yourself

Expand (x + 6)(x − 2). Write down all four partial products before you collect like terms. Then say what is missing from the answer x² − 12.

Flashcards

(14)

What is a binomial?

An expression with exactly two unlike terms, such as 5 − 6x or x² − x.

Is x² + 3x + 2 a binomial? Is 5xy?

Neither: x² + 3x + 2 has three terms, and 5xy is a single term.

What is a partial product?

One of the multiplication results that lead to an overall multiplication, like 100, 30, 20 and 6 for 12 × 13.

How many partial products do you get when you multiply two binomials?

Four: each of the two terms in one bracket times each of the two terms in the other.

In an area model for two binomials, what does each cell show?

One partial product: its row's term times its column's term.

What is expanded form?

Writing the product as a sum of terms, like x² + 4x + 3, instead of as (x + 3)(x + 1).

Does the order of the brackets matter: (x + 3)(x + 1) or (x + 1)(x + 3)?

No. Multiplication is commutative, so both expand to x² + 4x + 3.

How do you treat a subtraction like x − 2 when multiplying brackets?

As adding a negative: x + (−2).

How do you expand (x + 7)²?

Write it as (x + 7)(x + 7), then find all four partial products: x² + 14x + 49.

What is 4x × 2x?

8x². Multiply the numbers (4 × 2) and then the letters (x × x).

What is (a + b)(a − b), and what is it called?

a² − b², the difference of two squares.

Why do the middle terms vanish in (x + 7)(x − 7)?

+7x and −7x are a zero pair, so they cancel: the answer is x² − 49.

Which term usually comes first in the expanded answer?

The one with the largest exponent: x² terms, then x terms, then the number.

When does 'add the constants, multiply the constants' work?

For (x + a)(x + b), such as (x + 5)(x + 4). It doesn't work for products like (2x + 4)(2x + 5).

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Walk every path through (x + 10)(x − 15)

Build the product in a grid

No grid? Split it up

Squaring a binomial: find the slip

Difference of two squares, or not?

Is there a shortcut?

More KS3 Maths topics