Factor Theorem (Statement) MCQs Quiz | Class 9

This Class IX Mathematics (Code 041) quiz focuses on Unit II: Algebra. It specifically covers the statement of the Factor Theorem: (x-a) is a factor of polynomial P(x) if and only if P(a)=0. Complete the 10 MCQs below to test your understanding, submit to see your score, and download the answer PDF for your records.

Overview of Factor Theorem

The Factor Theorem is a fundamental concept in algebra for Class 9 students. It provides a direct link between the zeros (roots) of a polynomial and its factors. Essentially, it helps us determine if a linear polynomial is a divisor of a higher-degree polynomial without performing long division.

The Core Statement

If P(x) is a polynomial of degree n >= 1 and ‘a’ is any real number, then:

  • (x – a) is a factor of P(x) if P(a) = 0.
  • Conversely, if (x – a) is a factor of P(x), then P(a) = 0.

Understanding Through Cases

The sign inside the factor is crucial. Here is a quick reference table:

Linear Factor Zero to Check Condition for being a Factor
(x – a) x = a P(a) = 0
(x + a) x = -a P(-a) = 0
(ax – b) x = b/a P(b/a) = 0

Relationship with Remainder Theorem

The Factor Theorem is actually a special case of the Remainder Theorem. The Remainder Theorem states that when P(x) is divided by (x – a), the remainder is P(a). If this remainder P(a) is zero, then (x – a) divides P(x) completely, making it a factor.

Extra Practice Questions

  1. Check if (x – 1) is a factor of x^3 – 1. (Hint: Calculate P(1))
  2. Find the value of k if (x + 2) is a factor of x^2 + kx + 4.
  3. Is (x + 1) a factor of x^4 + x^3 + x^2 + 1?
  4. If P(a) = 0, what can you say about the polynomial P(x)?
  5. Use the theorem to determine if x – 3 is a factor of x^2 – 5x + 6.