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Class 9 Remainder Theorem — practice questions with answers

Polynomials · 5 sample questions from the school syllabus. Try each one, then open the answer and the worked solution.

1. By the Remainder Theorem, the remainder when a polynomial p(x) is divided by (x - a) equals which value?
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Answer: A) p(a)
The Remainder Theorem states that if p(x) is divided by (x - a), the remainder is p(a).
2. Find the remainder when p(x) = 4x³ - 12x² + 14x - 3 is divided by (2x - 1).
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Answer: A)
2x - 1 = 0 gives x = . p() = 4() - 12() + 14() - 3 = 0.5 - 3 + 7 - 3 = 1.5 = .
3. A shopkeeper in Jaipur models his profit (in Rs hundreds) by p(x) = x³ - 4x² + 5x - 2, where x is the week number.
Using the Remainder Theorem, find the remainder when p(x) is divided by (x - 1).
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Answer: D) 0
p(1) = 1 - 4 + 5 - 2 = 0. The remainder is 0.
4. The remainder when p(x) = kx³ + 3x² - 3 is divided by (x - 2) equals the remainder when q(x) = 2x³ + 5x + 7k is divided by (x - 2).
Find k.
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Answer: A) 17
p(2) = 8k + 12 - 3 = 8k + 9. q(2) = 16 + 10 + 7k = 7k + 26. Set equal: 8k + 9 = 7k + 26, so k = 17.
5. Final challenge: p(x) = 2x³ + 3x² - 11x - 6.
A student finds the remainders on division by (x - 2), (x + 3) and (x + ) and notices a pattern.
How many of these three divisors leave remainder 0 (i.e. are factors of p(x))?
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Answer: A) 3
p(2) = 16 + 12 - 22 - 6 = 0. p(-3) = -54 + 27 + 33 - 6 = 0. p() = 2() + 3() - 11() - 6 = . All three give remainder 0.
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