Multiplying a Binomial by a Trinomial — Grade 10 (CAPS)

Expanding products like (2x + 3)(x² − x + 4) reliably using a grid, then collecting like terms — plus the cube identities that come from it.

Grade 9 multiplied two binomials, giving four products. Grade 10 adds a third term to the second bracket — now there are six, and the usual mistake is simply losing one.

The rule: multiply every term in the first bracket by every term in the second bracket, then collect like terms. A binomial has 2 terms and a trinomial has 3, so there are always 2 × 3 = 6 products before you collect. Count them: if you have only five, one is missing.

1Use a grid so nothing goes missing

Each row-term multiplies each column-term. Six cells, then collect like terms.

Worked ExampleExpand: (2x + 3)(x² − x + 4)

1
2x × x² = 2x³; 2x × (−x) = −2x²; 2x × 4 = 8x.
First bracket term across the whole second bracket.
2
3 × x² = 3x²; 3 × (−x) = −3x; 3 × 4 = 12.
Then the second term.
3
Collect: 2x³ + (−2 + 3)x² + (8 − 3)x + 12.
Group like powers.
4
= 2x³ + x² + 5x + 12.

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Signs travel with their term. In (2x − 3)(x² + x − 5) the second row is −3 times each term, so −3 × (−5) = +15.

2Method 2: one term at a time

Once the grid feels natural, you can skip drawing it: multiply the whole trinomial by the first term of the binomial, then by the second term, and write the two rows one under the other.

Worked ExampleExpand: (x − 4)(2x² + 3x − 1)

1
x(2x² + 3x − 1) = 2x³ + 3x² − x.
First term across the trinomial.
2
−4(2x² + 3x − 1) = −8x² − 12x + 4.
The −4 carries its sign into every product.
3
Collect: 2x³ + (3 − 8)x² + (−1 − 12)x + 4.
Like powers together.
4
= 2x³ − 5x² − 13x + 4.

⚠️

x × x² = x³ (add the exponents), not x² and not 2x². With coefficients, multiply the numbers and add the exponents: 2x × 3x² = 6x³.

3Brackets with two variables

Worked ExampleExpand: (a − 2b)(a² + 3ab − b²)

1
a³ + 3a²b − ab²
a times each term.
2
−2a²b − 6ab² + 2b³
−2b times each term.
3
= a³ + a²b − 7ab² + 2b³.
a²b and ab² are different terms: only collect terms with the same letters to the same powers.

4Check your answer with a number

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Substitute a small value, such as x = 2, into the question and into your answer. For (2x + 3)(x² − x + 4): 7 × 6 = 42. For 2x³ + x² + 5x + 12: 16 + 4 + 10 + 12 = 42. They agree, so the expansion is almost certainly right; if they disagree, look for the slip. Avoid x = 0 and x = 1, which hide mistakes in the powers.

5Exam style: expand and simplify

Worked ExampleSimplify: (x − 1)(x² + x + 1) − (x + 2)(x − 3)

1
(x − 1)(x² + x + 1) = x³ + x² + x − x² − x − 1 = x³ − 1.
The middle terms cancel.
2
(x + 2)(x − 3) = x² − x − 6.
An ordinary product of binomials.
3
x³ − 1 − (x² − x − 6) = x³ − 1 − x² + x + 6.
Keep the second product in brackets until the minus has changed every sign.
4
= x³ − x² + x + 5.

6Find one coefficient without expanding everything

Multiple-choice and short questions often ask for just one term of the answer. You do not need all six products: pick out only the pairs whose powers add up to the power you want, and ignore the rest.

Worked ExampleFind the coefficient of x in (3x − 2)(x² + 4x − 5)

1
3x × (−5) = −15x.
An x from the first bracket times a constant from the second.
2
−2 × 4x = −8x.
A constant from the first bracket times an x from the second.
3
−15x − 8x = −23x, so the coefficient is −23.
No other pair of terms gives a power of 1.

7Where the cube identities come from

Worked ExampleExpand: (x + 2)(x² − 2x + 4)

1
x³ − 2x² + 4x + 2x² − 4x + 8.
Six products as usual.
2
The middle terms cancel in pairs: −2x² + 2x² = 0 and 4x − 4x = 0.
This is not a coincidence.
3
= x³ + 8.
A sum of two cubes — this is the identity you will factorise backwards next.

(a + b)(a² − ab + b²) = a³ + b³
(a − b)(a² + ab + b²) = a³ − b³

These two identities are why this lesson matters beyond expanding. Read backwards, they are how you factorise a sum or a difference of two cubes later in Grade 10: x³ + 8 = (x + 2)(x² − 2x + 4), exactly the product expanded above.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Expand: (x + 1)(x² + 2x + 3)
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x³ + 3x² + 5x + 3
2
Expand: (x + 3)(x² − x + 2)
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x³ + 2x² − x + 6
3
Expand: (2x + 1)(x² + 3x − 2)
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2x³ + 7x² − x − 2
4
Expand: (x − 2)(x² + 2x + 4)
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x³ − 8 (difference of cubes)
5
Expand: (x + 5)(x² − 5x + 25)
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x³ + 125
6
Expand: (3x − 1)(x² − x + 2)
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3x³ − 4x² + 7x − 2
7
How many products before collecting, for a binomial × trinomial?
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Six (2 × 3)