Factor Theorem (Proof) MCQs Quiz | Class 9

This theory-based quiz covers Class IX Mathematics (Code 041), Unit II: Algebra, specifically focusing on the Proof steps and logic of the Factor Theorem. It tests your understanding of the relationship between roots, factors, and the Remainder Theorem. Click Submit to verify your answers and download the PDF solution sheet.

Educational Content: Factor Theorem Proof & Logic

The Factor Theorem is a fundamental concept in algebra that links the roots of a polynomial to its linear factors. It is a direct consequence of the Remainder Theorem.

1. Statement of the Factor Theorem

If p(x) is a polynomial of degree n greater than or equal to 1 and ‘a’ is any real number, then:

  • (i) If p(a) = 0, then (x – a) is a factor of p(x).
  • (ii) If (x – a) is a factor of p(x), then p(a) = 0.

2. Logic of the Proof

The proof relies on the Division Algorithm: p(x) = (x – a)q(x) + p(a), where p(a) is the remainder.

  • Proof of part (i): If p(a) = 0, the equation becomes p(x) = (x – a)q(x). Since q(x) is a polynomial, this shows (x – a) divides p(x) perfectly. Thus, (x – a) is a factor.
  • Proof of part (ii): If (x – a) is a factor, then p(x) can be written as (x – a)g(x) for some polynomial g(x). Substituting x = a, we get p(a) = (a – a)g(a) = 0.

3. Comparison Table

ConceptConditionResult
Remainder TheoremDivide p(x) by (x – a)Remainder is p(a)
Factor TheoremIf p(a) = 0(x – a) is a factor
Root of EquationIf p(a) = 0x = a is a zero/root

4. Quick Revision Points

  • The degree of the polynomial must be at least 1.
  • The divisor is typically a linear polynomial of the form (x – a).
  • If divided by (x + a), check p(-a).
  • If divided by (ax – b), check p(b/a).

5. Extra Practice Questions

  1. Q: Check if (x-1) is a factor of p(x) = x^3 – 1. (Hint: Find p(1)).
  2. Q: Find k if (x-2) is a factor of x^2 + kx – 6.
  3. Q: Does the Factor Theorem apply to constant polynomials? (No, degree must be >= 1).
  4. Q: Show that x+2 is a factor of x^3 + 8.
  5. Q: If p(3) = 5, is (x-3) a factor of p(x)? (No, remainder is 5, not 0).