Polynomials & identities · Ages 13–14
Comparing coefficients
Expand, put in the same order, then match term by term.
How to do it
Method
Expand, put in the same order, then match term by term
(x + 2)(x + k) ≡ x2 + 7x + m: 2 + k = 7, k = 5, m = 10
Another way
Put in handy values
x = 0: 2k = m
In full
If ax2 + bx + c ≡ px2 + qx + r, the two sides are equal for every x, so like terms' coefficients must match: a = p, b = q, c = r. Expand one side, write both in the same order, then compare term by term. (x + 2)(x + k) ≡ x2 + 7x + m: the left is x2 + (2 + k)x + 2k, so comparing x's coefficients, 2 + k = 7 and k = 5; comparing the constants, m = 2k = 10. Or put in handy values: x = 0 gives 2k on the left and m on the right, so m = 2k; x = 1 gives 3(1 + k) = 8 + m; solve the two together. The answer is the same.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
If px2 + 2x + r ≡ 8x2 + qx + 8, then p = , q = and r = .
Fill in each blank
Show the answer and working
Answer: 8, 2, 8
In an identity, like terms' coefficients match: x2's, p = 8; x's, q = 2; the constants, r = 8.
Example 2
If (x + 9)(x + 7) ≡ x2 + px + q, then p = 16 and q = 16.
True or false?
Show the answer and working
Answer: False
The left is x2 + 16x + 63. Comparing coefficients: p = 16, q = 63.
Example 3
If 3x2 + bx + 6 ≡ (3x + 6)(x + 1), then b = .
Show the answer and working
Answer: 9
The right is 3x2 + 3x + 6x + 6 = 3x2 + 9x + 6. Comparing x's coefficients, b = 9.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.If px2 + 4x + r ≡ 5x2 + qx + 10, then p = , q = and r = .
Fill in each blank
Answer:
2.If (x + 5)(x + k) ≡ x2 + 6x + 5, then k = .
Answer:
3.If (x + 2)(x + k) ≡ x2 + mx − 12, then k = and m = .
Fill in each blank
Answer:
4.If (px + 7)(x + 6) ≡ 2x2 + qx + 42, what are p and q?
- A. p = 2, q = 13
- B. p = 42, q = 19
- C. p = 2, q = 20
- D. p = 2, q = 19
Choose the right answer
Answer:
5.If A(x + 7) + B(x − 7) ≡ −9x + 21, then A = and B = .
Fill in each blank
Answer:
All the answers
- 1. 5, 4, 10
- 2. 1
- 3. −6, −4
- 4. D. p = 2, q = 19
- 5. −3, −6
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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