Algebraic expressions · Ages 13–14

The cross method: leading coefficient 1 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.x² + 14x + 48 = (x + 6)(x + )
Answer:
2.x² + 5x + 4 = (x + a)(x + b), where a < b. a = and b = .
Fill in each blank
Answer:
3.x² + 12x + 27 = (x + 3)(x + )
Answer:
4.x² + x − 56 = (x + a)(x + b), where a < b. a = and b = .
Fill in each blank
Answer:
5.x² + 7x + 10 = (x + a)(x + b), where a < b. a = and b = .
Fill in each blank
Answer:
6.x² + 8xy + 7y² = (x + 7y)(x + y)
Answer:
7.x² + 16x + k factorises as (x + 8)(x + m), so k = .
Answer:
8.x² + 5xy + 4y² = (x + 4y)(x + y)
Answer:

13.x² + 14x + 13 can be written as two linear factors with whole-number coefficients multiplied.
True or false?
Working:
Answer:
14.A rectangle has area (x² + 5x + 4) cm²; its length and width are each x plus a positive whole number. The length is cm more than the width.
Working:
Answer:
15.Factorise x² + 9x + 8. Put the steps in order.
- A. Find two numbers: product = 8, sum = 9
- B. x² + 9x + 8 = (x + 1)(x + 8)
- C. Check: (x + 1)(x + 8) = x² + 8x + x + 8 = x² + 9x + 8
- D. 1 × 8 = 8, 1 + 8 = 9
Write the letters in the right order
Working:
Answer:
16.A rectangular flower bed has area (x² + 4x + 3) m² and length (x + 3) m. Its width is (x + ) m; when x = 7, the width is m.
Fill in each blank
Working:
Answer:
17.Factorise x² + 7xy + 6y². What do you get?
- A. (x + y)(x − 6y)
- B. (x − y)(x − 6y)
- C. (x + 1)(x + 6)y²
- D. (x + y)(x + 6y)
Choose the right answer
Working:
Answer:
18.To factorise x² + 2x − 24, find two numbers whose product is and sum is : they are and (the smaller first).
Fill in each blank
Working:
Answer:

Show the answers and working

The cross method: leading coefficient 1 worksheet: Answers and working

1. 8
Two numbers with product 48 and sum 14: 6 and 8.
2. 1, 4
Find two numbers with product 4 and sum 5: 1 × 4 = 4 and 1 + 4 = 5.
3. 9
Two numbers with product 27 and sum 12: 3 and 9.
4. −7, 8
Two numbers with product −56 and sum 1: (−7) × (8) = −56 and (−7) + (8) = 1.
5. 2, 5
Two numbers with product 10 and sum 7: (2) × (5) = 10 and (2) + (5) = 7.
6. 1
Find the two numbers as for x² + 8x + 7: 7 and 1, each with a y.
7. 64
8 + m = 16, so m = 8; k is their product, 8 × 8 = 64.
8. 1
Find the two numbers as for x² + 5x + 4: 4 and 1, each with a y.

9. A. (x + 2)(x + 6)
2 and 6 multiply to 12 and add to 8: (x + 2)(x + 6).
10. D. (x + 1)(x + 6)
1 and 6 multiply to 6 and add to 7: (x + 1)(x + 6).
11. False
The two numbers with product −6 and sum 5 are 6 and −1, so it is (x + 6)(x − 1).
12. A. (x + 1)(x − 10)
Product −10, sum −9: 1 and −10, so it is (x + 1)(x − 10).

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

13. True
13 is prime: the only positive whole numbers multiplying to 13 are 1 and 13, adding to 14. Only x² + 14x + 13 = (x + 1)(x + 13) factorises.
14. 3
x² + 5x + 4 = (x + 1)(x + 4), so they differ by 4 − 1 = 3 cm.
15. A → D → B → C
Say what is needed, find the two numbers, write the two factors, then expand to check.
16. 1, 8
1 and 3 multiply to 3 and add to 4: area = (x + 1)(x + 3), so the width is (x + 1) m, which is 8 m when x = 7.
17. D. (x + y)(x + 6y)
As with x² + 7x + 6, the numbers are 1 and 6, each with a y: (x + y)(x + 6y).
18. −24, 2, −4, 6
Product −24, sum 2: (−4) × (6) = −24 and (−4) + (6) = 2, so x² + 2x − 24 = (x + (−4))(x + (6)).

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