NCERT Chapter 2 MCQs – Remainder Theorem & Factorisation of Polynomials

Class: 9 | Subject: Mathematics | Chapter 2: Polynomials

✔ Based strictly on NCERT (Latest CBSE Syllabus)
✔ Covers Remainder Theorem & Factorisation Methods
✔ Ideal for School Exams & CBSE Standard Preparation

1. According to the Remainder Theorem, the remainder when p(x) is divided by (x − a) is:

A) p(a)
B) p(−a)
C) a
D) 0

Correct Answer: A) p(a)

Explanation: Remainder Theorem states that when a polynomial p(x) is divided by (x−a), the remainder is p(a).

2. If p(2)=0, then (x−2) is:

A) Remainder
B) Factor
C) Coefficient
D) Constant

Correct Answer: B) Factor

Explanation: If p(a)=0, then (x−a) is a factor of the polynomial.

3. Find remainder when p(x)=x²+3x+2 is divided by (x−1).

A) 6
B) 4
C) 2
D) 0

Correct Answer: A) 6

Explanation: By Remainder Theorem, p(1)=1+3+2=6.

4. Which of the following is a factor of x²−9?

A) x−3
B) x−4
C) x+5
D) x−1

Correct Answer: A) x−3

Explanation: x²−9=(x−3)(x+3).

5. If remainder is zero, the divisor is called:

A) Multiple
B) Factor
C) Constant
D) Zero

Correct Answer: B) Factor

Explanation: Zero remainder means exact division → divisor is a factor.

6. Factorise x²+5x+6:

A) (x+2)(x+3)
B) (x+1)(x+6)
C) (x+4)(x+2)
D) (x−2)(x−3)

Correct Answer: A)

Explanation: Numbers whose product=6 and sum=5 → 2 and 3.

7. The remainder when p(x)=2x³+3x²−x+1 is divided by (x−1) is:

A) 5
B) 3
C) 2
D) 0

Correct Answer: A)

Explanation: p(1)=2+3−1+1=5.

8. Factorise x²−5x+6:

A) (x−2)(x−3)
B) (x+2)(x+3)
C) (x−1)(x−6)
D) (x+1)(x−6)

Correct Answer: A)

Explanation: Product=6, sum=−5 → −2, −3.

9. If p(−1)=0, then factor is:

A) x+1
B) x−1
C) x−2
D) x+2

Correct Answer: A)

Explanation: p(a)=0 ⇒ (x−a) factor → a=−1 ⇒ x+1.

10. Factorise x³−x:

A) x(x−1)(x+1)
B) x(x−1)
C) x(x+1)
D) None

Correct Answer: A)

Explanation: x³−x=x(x²−1)=x(x−1)(x+1).

11. Remainder when divided by (x+2) is found by substituting:

A) 2
B) −2
C) 0
D) 1

Correct Answer: B)

Explanation: x+2 = x−(−2).

12. Factorise x²−16:

A) (x−4)(x+4)
B) (x−8)(x+2)
C) (x−2)(x+8)
D) None

Correct Answer: A)

Explanation: Difference of squares.

13. If p(3)=5, remainder when divided by (x−3) is:

A) 3
B) 5
C) 0
D) 8

Correct Answer: B)

Explanation: Remainder = p(3).

14. Factorise 2x²+7x+3:

A) (2x+1)(x+3)
B) (2x+3)(x+1)
C) (x+1)(x+3)
D) None

Correct Answer: A)

Explanation: Split middle term.

15. Remainder when p(x)=x³+1 divided by (x+1) is:

A) 2
B) 0
C) 1
D) −1

Correct Answer: B)

Explanation: p(−1)=−1+1=0.

16. Factorise x²−1:

A) (x−1)(x+1)
B) (x−1)²
C) (x+1)²
D) None

Correct Answer: A)

Explanation: Difference of squares.

17. If (x−4) is factor, then p(4)=

A) 4
B) 0
C) 1
D) −4

Correct Answer: B)

Explanation: Factor Theorem.

18. Factorise x²+4x+4:

A) (x+2)²
B) (x−2)²
C) (x+4)(x+1)
D) None

Correct Answer: A)

Explanation: Perfect square.

19. Remainder when divided by (x−0) is:

A) p(0)
B) 0
C) 1
D) x

Correct Answer: A)

Explanation: Substitute x=0.

20. Factorise x³+8:

A) (x+2)(x²−2x+4)
B) (x−2)(x²+2x+4)
C) (x+2)³
D) None

Correct Answer: A)

Explanation: Sum of cubes formula.

NCERT Chapter 2 MCQs – Remainder Theorem & Factorisation of Polynomials

NCERT Chapter 2 MCQs – Remainder Theorem & Factorisation of Polynomials

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