−6x(x − 1) ≡ px^2 + qx, so p = □ and q = □. Fill in each blank

−6, 6. The term outside times every term inside: p = (−6) × (1) = −6, q = (−6) × (−1) = 6. Same signs make a positive, different signs a negative.

After simplifying 3x(2x + 2) − 25x^2, the coefficient of x^2 is □.

−19. 3x × 2x = 6x^2, then take away 25x^2: 6 − 25 = −19.

−7x(7x − 2) = ? A. 49x^2 − 14x B. −49x^2 + 2 C. −49x^2 − 14x D. −49x^2 + 14x Choose the right answer

D. −49x^2 + 14x. −7x times each term: (−7x) × 7x = −49x^2 and (−7x) × (−2) = +14x.

Polynomials & identities · Ages 13–14

A monomial times a polynomial

The term outside times every term inside; same letters, add the indices.

How to do it

Method

The term outside times every term inside; same letters, add the indices

3x(2x − 5) = 6x² − 15x

Another way

A rectangle's area

3x × 2x = 6x², 3x × 5 = 15x

In full

Use the distributive law: the monomial outside multiplies every term inside the bracket. Multiplying monomials, multiply the coefficients and add the indices of the same letter: 3x × 2x = 6x², x² × x³ = x⁵. 3x(2x − 5) = 3x × 2x + 3x × (−5) = 6x² − 15x. With a negative outside, every term changes sign: −2x(x − 4) = −2x² + 8x; the usual slip is forgetting that (−2x) × (−4) is positive. Or draw a rectangle 3x wide, its length split into 2x and 5: the two areas, 6x² and 15x, add up to the product. The answer is the same.

Worked examples

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

Example 1
−6x(x − 1) ≡ px² + qx, so p = and q = .
Fill in each blank
Show the answer and working
Answer: −6, 6
The term outside times every term inside: p = (−6) × (1) = −6, q = (−6) × (−1) = 6. Same signs make a positive, different signs a negative.
Example 2
After simplifying 3x(2x + 2) − 25x², the coefficient of x² is .
Show the answer and working
Answer: −19
3x × 2x = 6x², then take away 25x²: 6 − 25 = −19.
Example 3
−7x(7x − 2) = ?
- A. 49x² − 14x
- B. −49x² + 2
- C. −49x² − 14x
- D. −49x² + 14x
Choose the right answer
Show the answer and working
Answer: D. −49x² + 14x
−7x times each term: (−7x) × 7x = −49x² and (−7x) × (−2) = +14x.

Practice questions to try

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

1.4x² × 4x³ = ?
- A. 16x⁶
- B. 8x⁶
- C. 8x⁵
- D. 16x⁵
Choose the right answer
Answer:
2.3x(2x + 18) = x² + x
Fill in each blank
Answer:
3.−3x(6x − 4) = ?
- A. 18x² − 12x
- B. −18x² + 4
- C. −18x² + 12x
- D. −18x² − 12x
Choose the right answer
Answer:
4.After simplifying 9x(8x + 8) − 20x², the coefficient of x² is .
Answer:
5.7x²y × 7xy⁵ = 49x³y⁶
True or false?
Answer:

All the answers

1. D. 16x⁵
2. 6, 54
3. C. −18x² + 12x
4. 52
5. True

Free worksheet: A monomial times a polynomialWorking 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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