(−9x^2 − 2x) − (−9x^2 + 5x) ≡ px^2 + qx, so p = □ and q = □. Fill in each blank

0, −7. Taking the bracket away changes every sign: p = (−9) − (−9) = 0, q = (−2) − (5) = −7.

(9x − 1) + (4x + 7) − (4x − 2) ≡ px + q, so p = □ and q = □. Fill in each blank

9, 8. Copy the bracket added; change every sign in the one taken away: p = 9 + 4 − 4 = 9, q = (−1) + (7) − (−2) = 8.

Take 3x^2 + 8x − 2 away from 9x^2 − 4x + 7. What do you get? A. 6x^2 − 12x + 9 B. 12x^2 + 4x + 5 C. 6x^2 + 4x + 9 D. −6x^2 + 12x − 9 Choose the right answer

A. 6x^2 − 12x + 9. A − B = 9x^2 − 4x + 7 − 3x^2 − 8x + 2; combining like terms gives 6x^2 − 12x + 9.

Polynomials & identities · Ages 13–14

Adding and subtracting polynomials

Taking a bracket away changes every sign inside it.

How to do it

Method

Taking a bracket away changes every sign inside it

(3x² − 2x + 5) − (x² − 4x + 1) = 2x² + 2x + 4

Another way

Line up like terms in columns

x² under x², x under x, number under number

In full

Only like terms can be combined: the same letters, each to the same index (3x² and −5x² are like terms; x² and x are not). Combine them by adding the coefficients and keeping the letters: 3x² − 5x² = −2x². Adding a bracket, copy what is inside; taking a bracket away, change the sign of every term inside: (3x² − 2x + 5) − (x² − 4x + 1) = 3x² − 2x + 5 − x² + 4x − 1 = 2x² + 2x + 4. The commonest slip is changing only the first sign. Or set them out in columns, like terms under each other, and add or subtract down each column. The answer is the same.

Worked examples

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

Example 1
(−9x² − 2x) − (−9x² + 5x) ≡ px² + qx, so p = and q = .
Fill in each blank
Show the answer and working
Answer: 0, −7
Taking the bracket away changes every sign: p = (−9) − (−9) = 0, q = (−2) − (5) = −7.
Example 2
(9x − 1) + (4x + 7) − (4x − 2) ≡ px + q, so p = and q = .
Fill in each blank
Show the answer and working
Answer: 9, 8
Copy the bracket added; change every sign in the one taken away: p = 9 + 4 − 4 = 9, q = (−1) + (7) − (−2) = 8.
Example 3
Take 3x² + 8x − 2 away from 9x² − 4x + 7. What do you get?
- A. 6x² − 12x + 9
- B. 12x² + 4x + 5
- C. 6x² + 4x + 9
- D. −6x² + 12x − 9
Choose the right answer
Show the answer and working
Answer: A. 6x² − 12x + 9
A − B = 9x² − 4x + 7 − 3x² − 8x + 2; combining like terms gives 6x² − 12x + 9.

Practice questions to try

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

1.(2x + 9) + (4x + 8) = x +
Fill in each blank
Answer:
2.Simplify (5x² + 2x + 17) − (2x² + 5x + 11). What do you get?
- A. 3x² + 7x + 28
- B. 3x² − 3x + 28
- C. 3x² − 3x + 6
- D. 7x² + 7x + 28
Choose the right answer
Answer:
3.After simplifying 2x²y − 8xy² + 6x²y + 3xy², the coefficient of xy² is .
Answer:
4.(−9x² − 4x) − (−3x² + 6x) ≡ px² + qx, so p = and q = .
Fill in each blank
Answer:
5.Take x² + 9x − 19 away from 2x² − 7x + 9. What do you get?
- A. 3x² + 2x − 10
- B. −x² + 16x − 28
- C. x² − 16x + 28
- D. x² + 2x + 28
Choose the right answer
Answer:

All the answers

1. 6, 17
2. C. 3x² − 3x + 6
3. −5
4. −6, −10
5. C. x² − 16x + 28

Free worksheet: Adding and subtracting 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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