Single correct option
When a polynomial p(x) is divided by (x − 1) the remainder is 5, and when it is divided by (x + 2) the remainder is −4. What is the remainder when p(x) is divided by (x − 1)(x + 2)?
- A. 3x + 2
- B. 2x + 3
- C. x + 4
- D. 3x − 2
Show answer and solution
Answer: (A) 3x + 2
The divisor has degree 2, so the remainder has the form ax + b: p(x) = (x − 1)(x + 2)q(x) + ax + b.
By the remainder theorem:
p(1) = a + b = 5
p(−2) = −2a + b = −4
Subtracting: 3a = 9 → a = 3, b = 2. Remainder = 3x + 2.
Check the options quickly: 2x + 3 gives 5 at x = 1 but −1 at x = −2 ✗; x + 4 gives 5 and 2 ✗; 3x − 2 gives 1 at x = 1 ✗.
By the remainder theorem:
p(1) = a + b = 5
p(−2) = −2a + b = −4
Subtracting: 3a = 9 → a = 3, b = 2. Remainder = 3x + 2.
Check the options quickly: 2x + 3 gives 5 at x = 1 but −1 at x = −2 ✗; x + 4 gives 5 and 2 ✗; 3x − 2 gives 1 at x = 1 ✗.
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