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