Let p(x) = ax + b and q(x) = cx + d be two linear polynomials such that:
(i) p(0) = 5.
(ii) The polynomial p(x) − q(x) cuts the x-axis at (3, 0).
(iii) The sum p(x) + q(x) is equal to 6x + 4 for all real x.
Find the polynomials p(x) and q(x).
Step-by-step solution
To find: a, b, c, d
Idea: Use each condition in turn. “Equal for all real x” means the coefficients of x match and the constants match.
- (i) p(0) = b = 5.½ mark
- (iii) (a + c)x + (b + d) = 6x + 4 for all x, so a + c = 6 and b + d = 4. Hence d = 4 − 5 = −1.1 mark
- (ii) p(x) − q(x) = (a − c)x + (b − d) = (a − c)x + 6. It is 0 at x = 3: 3(a − c) + 6 = 0, so a − c = −2.1 mark
- Solve a + c = 6 and a − c = −2: adding, 2a = 4, so a = 2 and c = 4.1 mark
- So p(x) = 2x + 5 and q(x) = 4x − 1.½ mark
Check: p(0) = 5 ✓. p(x) − q(x) = −2x + 6, which is 0 at x = 3 ✓. p(x) + q(x) = 6x + 4 ✓.
Answer to write in the exam
p(0) = 5 ⇒ b = 5
p(x) + q(x) = (a + c)x + (b + d) = 6x + 4 ⇒ a + c = 6, b + d = 4 ⇒ d = −1
p(3) − q(3) = 0 ⇒ 3(a − c) + (b − d) = 0 ⇒ 3(a − c) + 6 = 0 ⇒ a − c = −2
a + c = 6, a − c = −2 ⇒ a = 2, c = 4
∴ p(x) = 2x + 5, q(x) = 4x − 1
Common mistakes that cost marks
- Writing b − d = 5 − 1 = 4 instead of 5 − (−1) = 6.
- Using only a + c = 6 and guessing a = c = 3. Condition (ii) is needed to separate a and c.
- Reading “cuts the x-axis at (3, 0)” as p(0) − q(0) = 3. It means the value is 0 when x = 3.
How this can come in the exam
If p(x) = ax + b and p(x) + (2x − 3) = 7x + 1 for all x, then p(x) =
- 5x + 4
- 9x − 2
- 5x − 2
- 7x + 4
Show answer
(A) 5x + 4
a + 2 = 7 ⇒ a = 5; b − 3 = 1 ⇒ b = 4.
p(x) = ax + b and q(x) = cx + d satisfy q(0) = 1, p(x) + q(x) = 5x + 3 and p(x) − q(x) = x + 1. Find p(x) and q(x).
Show answer
d = 1 and b + d = 3 ⇒ b = 2 (1 mark). a + c = 5 and a − c = 1 ⇒ a = 3, c = 2 (1 mark). p(x) = 3x + 2, q(x) = 2x + 1 (and b − d = 1 ✓) (1 mark).Try one yourself
Find linear p(x) and q(x) with p(0) = 3, p(x) + q(x) = 4x + 1, and p(x) − q(x) zero at x = 1.
Show answer
b = 3, d = −2; a + c = 4 and (a − c) + 5 = 0 ⇒ a − c = −5 ⇒ a = −12, c = 92. p(x) = −12x + 3, q(x) = 92x − 2.
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