# Polynomials

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## Remainders

4x3 + ax2 – bx + 3 divided by x – 2 leaves remainder 2, divided by x + 3 leaves remainder 3. Find remainder when it is divided by x + 2.
1. 26.8
2. 29.2
3. 32.2
4. 35.2

• Correct Answer
Choice D. 35.2

## Explanatory Answer

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## Detailed Solution

Let f(x) = 4x3 + ax2 – bx + 3.
Remainder on division of f(x) by x - 2 = f(2)
Or, f(2) = 32 + 4a – 2b + 3 = 2
4a – 2b = -33 ----Eqn (1)
Remainder on division of f(x) by x + 3 = f(-3)
Or, f(-3) = –108 + 9a + 3b + 3 = 3
9a + 3b = 108
Dividing by 3, the above equation becomes
3a + b = 36. Multiplying this equation by 2, we get
6a + 2b = 72 ----Eqn (2)
Adding equations (1) and (2), we have 10a = 39
a = 3.9
4(3.9) – 2b = –33
b = 24.3
f(x) = 4x3 + 3.9x2 – 24.3x + 3
Remainder on division of f(x) by x + 2 = f(-2)
f(–2) = 4(–8) + 3.9(4) – 24.3(–2) + 3 = 35.2
Answer choice (d)

Correct Answer: 35.2.

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