Multiplying a Binomial by a Trinomial (Grade 10)

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.

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.

⚠️

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

2Where 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

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