Polynomials — Previous Year Questions (Class 9 Mathematics)
Polynomials are algebraic expressions with variables and coefficients. Learn to identify polynomial types, find zeros, and apply the Remainder and Factor T
TL;DR: Polynomials are algebraic expressions with variables and coefficients. Learn to identify polynomial types, find zeros, and apply the Remainder and Fac…
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Polynomials are algebraic expressions with variables and coefficients. Learn to identify polynomial types, find zeros, and apply the Remainder and Factor T
Polynomials — Previous Year Questions with Solutions
Q (2023, 5 marks): If x = 2 is a zero of the polynomial p(x) = x^3 - 2x^2 - 5x + 6, find the other two zeros.
Answer: Since x = 2 is a zero, (x - 2) is a factor.
Divide p(x) by (x - 2):
x^3 - 2x^2 - 5x + 6 = (x - 2)(x^2 - 5)
For x^2 - 5 = 0:
x^2 = 5
x = ±sqrt(5)
The three zeros are: 2, sqrt(5), -sqrt(5)
Q (2022, 2 marks): Find the remainder when p(x) = x^3 + 3x^2 - 5x + 4 is divided by (x - 2).
Answer: Using Remainder Theorem: remainder = p(2)
p(2) = (2)^3 + 3(2)^2 - 5(2) + 4
= 8 + 12 - 10 + 4
= 14
Q (2023, 3 marks): Factorize: 2x^3 + 3x^2 - 8x - 12
Answer: Group terms: (2x^3 - 8x) + (3x^2 - 12)
= 2x(x^2 - 4) + 3(x^2 - 4)
= (x^2 - 4)(2x + 3)
= (x - 2)(x + 2)(2x + 3)
Q (2021, 3 marks): Divide p(x) = x^3 - 6x^2 + 11x - 6 by (x - 1) and find quotient and remainder.
Answer: Using polynomial division:
x^3 - 6x^2 + 11x - 6 = (x - 1)(x^2 - 5x + 6) + 0
Quotient: x^2 - 5x + 6
Remainder: 0
(This factors as (x - 1)(x - 2)(x - 3))
Q (2022, 3 marks): If p(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, show that x = 1 is a zero of multiplicity 4.
Answer: p(1) = 1 - 4 + 6 - 4 + 1 = 0
Notice that p(x) = (x - 1)^4
Expanding: (x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1
Therefore, x = 1 is a zero with multiplicity 4
Q (2023, 3 marks): If the polynomial p(x) = x^2 + (a - 1)x + (b + 2) has zeros 2 and -3, find a and b.
Answer: Sum of zeros = 2 + (-3) = -1
Using Vieta's formula: -(a - 1) = -1
a - 1 = 1
a = 2
Product of zeros = 2 × (-3) = -6
Using Vieta's formula: b + 2 = -6
b = -8
Therefore: a = 2, b = -8
Frequently Asked Questions
What is the Remainder Theorem?
If a polynomial p(x) is divided by (x - a), the remainder is p(a). This helps find remainders without performing full polynomial division.
What is the Fundamental Theorem of Algebra?
A polynomial of degree n has exactly n roots (real or complex, counting multiplicities). This helps determine how many zeros a polynomial must have.
More Class 9 Mathematics PYQs
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