Algebraic expressions · Ages 13–14

The cross method: other leading coefficients worksheet

Part of: Factorisation

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Algebraic expressions · Ages 13–14

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.Factorising 3x² + 28x + 9 by the cross method, with 3x and x on the left and 1 and 9 on the right: across, 3x × 9 = x and x × 1 = x, which add to x.
Fill in each blank
Answer:
2.3x² + 19x + 28 = (3x + 7)(x + )
Answer:
3.5x² + 19x + 12 = (5x + 4)(x + )
Answer:
4.Factorising 4x² + 11x + 6 by the cross method, with 4x and x on the left and 3 and 2 on the right: across, 4x × 2 = x and x × 3 = x, which add to x.
Fill in each blank
Answer:
5.2x² − 15x + 27 ≡ (2x + h)(x + k), so h = and k = .
Fill in each blank
Answer:
6.2x² + 19x + 42 ≡ (2x + h)(x + k), so h = and k = .
Fill in each blank
Answer:
7.Splitting the middle term of 5x² + 39x + 54: 5 × 54 = ; split 39x into x + x (the larger first), so that the two coefficients multiply to 5 × 54.
Fill in each blank
Answer:
8.Splitting the middle term of 4x² + 19x + 21: 4 × 21 = ; split 19x into x + x (the larger first), so that the two coefficients multiply to 4 × 21.
Fill in each blank
Answer:

13.3x² + 11x + 8 = 3x² + x + 8x + 8 = 3x(x + ) + 8(x + 1) = (x + 1)(x + 8)
Fill in each blank
Working:
Answer:
14.Factorise 3x² − 22x − 45. What do you get?
- A. (3x + 5)(x + 9)
- B. (3x + 5)(x − 9)
- C. (3x − 5)(x + 9)
- D. (3x − 9)(x + 5)
Choose the right answer
Working:
Answer:
15.A rectangle has area (3x² + kx + 4) cm², length (3x + 1) cm and width (x + 4) cm. k = .
Working:
Answer:
16.(x + 1) is a factor of 2x² + 19x + 9.
True or false?
Working:
Answer:
17.Factorise 2x² + 7x + 6 by the cross method. Put the steps in order.
- A. Split the ends: 2 = 2 × 1, 6 = 3 × 2
- B. 2x² + 7x + 6 = (2x + 3)(x + 2)
- C. Across: 2x × 2 + x × 3 = 7x
Write the letters in the right order
Working:
Answer:
18.A rectangle has area (2x² + 19x + 42) m² and width (x + 6) m. Its length is (x + ) m.
Fill in each blank
Working:
Answer:

Show the answers and working

The cross method: other leading coefficients worksheet: Answers and working

1. 27, 1, 28
27x + x = 28x, the middle term itself, so 3x² + 28x + 9 = (3x + 1)(x + 9).
2. 4
7 × □ = 28, so □ = 4. Check across: 3x × 4 + x × 7 = 19x.
3. 3
4 × □ = 12, so □ = 3. Check across: 5x × 3 + x × 4 = 19x.
4. 8, 3, 11
8x + 3x = 11x, the middle term itself, so 4x² + 11x + 6 = (4x + 3)(x + 2).
5. −9, −3
hk = 27 and, across, 2k + h = −15: h = −9 and k = −3 fit both.
6. 7, 6
hk = 42 and, across, 2k + h = 19: h = 7 and k = 6 fit both.
7. 270, 30, 9
30 × 9 = 270 and 30 + 9 = 39. Split, then grouped in pairs, it gives (5x + 9)(x + 6).
8. 84, 12, 7
12 × 7 = 84 and 12 + 7 = 19. Split, then grouped in pairs, it gives (4x + 7)(x + 3).

9. D. (5x + 4)(x + 8)
5 = 5 × 1 and 32 = 4 × 8; across, 5x × 8 + x × 4 = 44x, so it is (5x + 4)(x + 8).
10. A. (3x + 7)(x + 4)
3 = 3 × 1 and 28 = 7 × 4; across, 3x × 4 + x × 7 = 19x, so it is (3x + 7)(x + 4).
11. True
Check across: 5x × 8 + x × 2 must be 42x. The right one is (5x + 2)(x + 8).
12. C. 3(3x + 4)(x + 2)
Take out the common factor 3: 3(3x² + 10x + 8); then the cross method gives 3(3x + 4)(x + 2).

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 3, 1, 3
3 × 8 = 3 × 8 and 3 + 8 = 11: split 11x into 3x + 8x, then group in pairs.
14. B. (3x + 5)(x − 9)
Across: 3x × (−9) + x × (5) = −22x, and the constant (5) × (−9) = −45.
15. 13
The x terms of (3x + 1)(x + 4) are 3x × 4 + 1 × x = 13x, so k = 13.
16. False
2x² + 19x + 9 = (2x + 1)(x + 9): its linear factors are 2x + 1 and x + 9.
17. A → C → B
Split 2 and 6 into factors in a cross; only when the cross products add to 7x is each row written as a factor.
18. 2, 7
By the cross method, 2x² + 19x + 42 = (2x + 7)(x + 6), since 2x × 6 + x × 7 = 19x. So the length is (2x + 7) m.

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