Class 9 Factor Theorem and Factorisation — 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 Factor Theorem, (x - a) is a factor of a polynomial p(x) if and only if which condition holds?
Ap(a) = 0
Bp(0) = a
Cp(a) = 1
Dp(-a) = 0
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Answer: A) p(a) = 0
Factor Theorem: (x - a) is a factor of p(x) exactly when the remainder p(a) equals 0.
2. If (x + 1) is a factor of p(x) = 2x³ + ax² + 2x - 1, find a.
A4
B10
C6
D5
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Answer: D) 5
p(-1) = 0: 2(-1)³ + a(-1)² + 2(-1) - 1 = -2 + a - 2 - 1 = a - 5 = 0, so a = 5.
3. If (2x - 3) is a factor of p(x), which value of x makes p(x) = 0?
A23
B32
C−23
D3
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Answer: A) 23
Set 2x - 3 = 0, so x=23. By the Factor Theorem p(23)=0.
4. When p(x) = x³ - 3x² + 4x - k is divided by (x - 2), it leaves no remainder. Use the Factor Theorem to determine the value of k.
A5
B3
C8
D4
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Answer: D) 4
No remainder means p(2) = 0: 8 - 12 + 8 - k = 0, so 4 - k = 0, giving k = 4.
5. A challenge problem: the polynomial p(x) = x³ + px² + qx + r has roots 1, 2 and 3. Using the factored form (x - 1)(x - 2)(x - 3), expand and identify the constant term r. What is the value of r?
A-12
B-6
C-5
D-7
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Answer: B) -6
(x - 1)(x - 2)(x - 3) = x³ - 6x² + 11x - 6. The constant term is r = -6 (equal to -(1 x 2 x 3)).