Practice the questions given in the worksheet on Remainder Theorem.
1. Use the Remainder Theorem, find the remainder when 4x\(^{3}\)  3x\(^{2}\) + 2x  4 is divided by x + 1.
2. If p(y) = y\(^{3}\) + y\(^{2}\)  2y + 1, using Remainder Theorem, find the remainder, when p(y) is divided by (y – 3), find the value of p(a).
3. Find the remainder (without division) when
(a) x\(^{2}\)  2x + 4 is divided by x  1
(b) 2x\(^{3}\)  3x\(^{2}\) + 7x  8 is divided by x  1
4. Use the Remainder Theorem, find the remainder when x\(^{4}\)  3x\(^{2}\) + 4x  12 is divided by x  3.
5. Find the remainder (without division) when
(a) x\(^{3}\) + 4x + 2 is divisible by x + 2
(b) 4x\(^{3}\)  3x\(^{2}\) + 5x + 4 is divided by 2x + 1
(c) 4x\(^{3}\) + 5x\(^{2}\) + 6x  7 is divided by 2x  1
6. What number should be added to x\(^{2}\) + 5 so that the resulting polynomial leaves the remainder 3 when divided by x + 3?
7. Use the Remainder Theorem, find the remainder when 4x\(^{3}\)  3x\(^{2}\) + 2x  4 is divided by x + 1.
8. What number should be subtracted from 3x\(^{2}\) + 5x so that the resulting polynomial leaves the remainder 1 when divided by 2x + 5?
9. Use the Remainder Theorem, find the remainder when x\(^{6}\) + 3x\(^{2}\) + 10 is divided by x  2.
10. Find a if the remainder is a when x\(^{3}\) + 3x\(^{2}\)  ax + 3 is divided by x  2.
11. If the polynomials ax\(^{3}\) + 4x\(^{2}\) + 3x – 4 and x\(^{3}\)  4x + a leave the same remainder when divided by (x  3), find the value of a.
12. Find the value of k if the remainder is 3 when kx\(^{3}\) + 8x\(^{2}\)  4x + 10 is divided by x +1.
13. If both ax\(^{3}\) + 2x\(^{2}\)  3 and x\(^{2}\)  ax + 4 leave the same remainder when divided by x  2, find a.
Answers for the worksheet on Remainder Theorem are given below:
Answers:
1. 13
2. 31, a\(^{3}\) + a\(^{2}\)  2a + 1
3. (a) 3
(b) 2
4. 54
5. (a) 14
(b) \(\frac{1}{4}\)
(c) \(\frac{9}{4}\)
6. 11
7. 13
8. \(\frac{21}{4}\)
9. 86
10. \(\frac{23}{3}\)
11. a = 1.
12. 25
13. \(\frac{3}{10}\)
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