(4x − 1)(3x + 5) ≡ px^2 + qx + r, so p = □, q = □ and r = □. Fill in each blank

12, 17, −5. p = 4 × 3 = 12; the x terms are two products added: (20) + (−3) = 17; r = (−1) × (5) = −5.

(x + 4)(x + 4) = x^2 + 8x + 8 True or false?

False. The constant is the two numbers multiplied: 4 × 4 = 16; it is x's coefficient that adds them: 4 + 4 = 8.

(x + 1)(x^2 + 3x + 1) ≡ x^3 + px^2 + qx + r, so p = □, q = □ and r = □. Fill in each blank

4, 4, 1. Two terms times three make 6 products: x^3, 3x^2, x, x^2, 3x and 1. p = 3 + 1 = 4, q = 1 + 3 = 4, r = 1.

Polynomials & identities · Ages 13–14

Multiplying polynomials

Every term times every term, then combine like terms.

How to do it

Method

Every term times every term, then combine like terms

(x + 3)(2x − 5) = 2x² + x − 15

Another way

The grid method

Cells 2x², −5x, 6x, −15, combined to 2x² + x − 15

In full

Every term of the first bracket multiplies every term of the second, then like terms are combined. (x + 3)(2x − 5) = x × 2x + x × (−5) + 3 × 2x + 3 × (−5) = 2x² − 5x + 6x − 15 = 2x² + x − 15. Two terms times two terms make 4 products, two times three make 6: count them and none is missed. Or use a grid: one bracket's terms along the top, the other's down the side, each cell the product, then add the cells that are like terms. The answer is the same. Check by putting in x = 1: (1 + 3)(2 − 5) = −12 and 2 + 1 − 15 = −12.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
(4x − 1)(3x + 5) ≡ px² + qx + r, so p = , q = and r = .
Fill in each blank
Show the answer and working
Answer: 12, 17, −5
p = 4 × 3 = 12; the x terms are two products added: (20) + (−3) = 17; r = (−1) × (5) = −5.
Example 2
(x + 4)(x + 4) = x² + 8x + 8
True or false?
Show the answer and working
Answer: False
The constant is the two numbers multiplied: 4 × 4 = 16; it is x's coefficient that adds them: 4 + 4 = 8.
Example 3
(x + 1)(x² + 3x + 1) ≡ x³ + px² + qx + r, so p = , q = and r = .
Fill in each blank
Show the answer and working
Answer: 4, 4, 1
Two terms times three make 6 products: x³, 3x², x, x², 3x and 1. p = 3 + 1 = 4, q = 1 + 3 = 4, r = 1.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.(x + 4)(x − 2) = ?
- A. x² + 2x − 8
- B. x² + 6x − 8
- C. x² − 8
- D. x² + 2x + 8
Choose the right answer
Answer:
2.(x + 5)(x + 2) = x² + 7x + 7
True or false?
Answer:
3.(3x + 5)(x + 7) ≡ px² + qx + r, so p = , q = and r = .
Fill in each blank
Answer:
4.(x + 7)(x² + 9x + 2) ≡ x³ + px² + qx + r, so p = , q = and r = .
Fill in each blank
Answer:
5.(4x + 9)(x − 4) = ?
- A. 4x² + 25x − 36
- B. 4x² − 7x − 36
- C. 4x² − 7x + 36
- D. 4x² − 36
Choose the right answer
Answer:

All the answers

1. A. x² + 2x − 8
2. False
3. 3, 26, 35
4. 16, 65, 14
5. B. 4x² − 7x − 36

Free worksheet: Multiplying polynomialsWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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