Algebraic expressions · Ages 13–14

Multiplying polynomials worksheet

Part of: Polynomials & identities

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Algebraic expressions · Ages 13–14

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Name:Date:Score:／48

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

1.(x + 5)(x + 9) = x² + x +
Fill in each blank
Answer:
2.(x + 8)(x + 3) = x² + x +
Fill in each blank
Answer:
3.(4x + 5)(2x − 8) ≡ px² + qx + r, so p = , q = and r = .
Fill in each blank
Answer:
4.Expanding (x + 9)(x² − x + 6), the coefficient of x is .
Answer:
5.(2x − 2)(2x + 9) ≡ px² + qx + r, so p = , q = and r = .
Fill in each blank
Answer:
6.Expanding (x + 6)(x² − 2x + 24), the coefficient of x is .
Answer:
7.(x + 1)(x² + 6x + 8) ≡ x³ + px² + qx + r, so p = , q = and r = .
Fill in each blank
Answer:
8.(x + 8)(x² + 6x + 4) ≡ x³ + px² + qx + r, so p = , q = and r = .
Fill in each blank
Answer:

13.A rectangular garden is (x + 8) m long and (x + 1) m wide. Its area is (x² + x + ) m².
Fill in each blank
Working:
Answer:
14.A rectangular sheet is (2x + 14) cm long and (x + 14) cm wide. A square of side 2 cm is cut from each corner. The area left is (x² + x + ) cm².
Fill in each blank
Working:
Answer:
15.A cuboid is (x + 7) cm long, (x − 9) cm wide and 2 cm high. What is its volume in cm³?
- A. 2x² + 32x − 126
- B. 2x² − 4x − 126
- C. 2x² − 2x − 63
- D. 2x² − 126
Choose the right answer
Working:
Answer:
16.Expand (x + 5)(2x − 7). Put the steps in order.
- A. = 2x² − 7x + 10x − 35
- B. Expression = x(2x − 7) + 5(2x − 7)
- C. = 2x² + 3x − 35
Write the letters in the right order
Working:
Answer:
17.Rectangle A is (x + 10) cm by (x + 10) cm; rectangle B is (x + 12) cm by (x + 7) cm. A's area is (x + ) cm² more than B's.
Fill in each blank
Working:
Answer:
18.A rectangular pool is 3x m long and x m wide, with a path 2 m wide all round it. The path's area is (x + ) m².
Fill in each blank
Working:
Answer:

Show the answers and working

Multiplying polynomials worksheet: Answers and working

1. 14, 45
Every term times every term: x² + 9x + 5x + 45, which combines to x² + 14x + 45.
2. 11, 24
Every term times every term: x² + 3x + 8x + 24, which combines to x² + 11x + 24.
3. 8, −22, −40
p = 4 × 2 = 8; the x terms are two products added: (−32) + (10) = −22; r = (5) × (−8) = −40.
4. −3
Only two products give an x term: x × 6 = 6x and 9 × (−x) = −9x, together 6 − 9 = −3.
5. 4, 14, −18
p = 2 × 2 = 4; the x terms are two products added: (18) + (−4) = 14; r = (−2) × (9) = −18.
6. 12
Only two products give an x term: x × 24 = 24x and 6 × (−2x) = −12x, together 24 − 12 = 12.
7. 7, 14, 8
Two terms times three make 6 products: x³, 6x², 8x, x², 6x and 8. p = 6 + 1 = 7, q = 8 + 6 = 14, r = 8.
8. 14, 52, 32
Two terms times three make 6 products: x³, 6x², 4x, 8x², 48x and 32. p = 6 + 8 = 14, q = 4 + 48 = 52, r = 32.

9. C. x² + 4x − 45
Every term times every term: x² − 5x + 9x − 45 = x² + 4x − 45.
10. True
The constant is the two numbers multiplied: 6 × 3 = 18; it is x's coefficient that adds them: 6 + 3 = 9.
11. A. 16x² + 8x − 24
Four products: 16x², −16x, 24x and −24; combining the middle two gives 16x² + 8x − 24.
12. A. 9x² − 21x − 8
Four products: 9x², −24x, 3x and −8; combining the middle two gives 9x² − 21x − 8.

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

13. 9, 8
Area = (x + 8)(x + 1) = x² + x + 8x + 8 = x² + 9x + 8.
14. 2, 42, 180
The sheet is (2x + 14)(x + 14) = 2x² + 42x + 196; take away four squares, 4 × 4: 2x² + 42x + 180.
15. B. 2x² − 4x − 126
Volume = 2(x + 7)(x − 9) = 2(x² − 2x − 63) = 2x² − 4x − 126.
16. B → A → C
Each term of the first bracket times the second bracket, multiplied out, then like terms combined.
17. 1, 16
(x + 10)(x + 10) − (x + 12)(x + 7): the x² cancel; x terms (10 + 10) − (12 + 7) = 1; numbers 10 × 10 − 12 × 7 = 16.
18. 16, 16
The pool with its path is (3x + 4)(x + 4); take away the pool, 3x²: the path is 16x + 16.

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