Question

Assertion (A) and Reason (R)

Assertion (A):
If a polynomial \(f(x)\) is divisible by \((x-a)^m\), but not divisible by \((x-a)^{m+1}\), then \(a\) is called a root of multiplicity \(m\) of the equation \(f(x)=0\).

Reason (R):
\(x=-1\) is the root of multiplicity \(3\) of the equation \[ x^4+x^3-3x^2-5x-2=0. \]
 

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

If \((x-a)^m\) is a factor of a polynomial, then \(a\) is a repeated root of multiplicity \(m\).

Answer 1

The Correct Option is B

Concept:
If \((x-a)^m\) divides \(f(x)\), but \((x-a)^{m+1}\) does not divide \(f(x)\), then \(a\) is a root of multiplicity \(m\).

Step 1: Check Assertion.
The assertion gives the correct definition of a root of multiplicity \(m\). \[ A \text{ is correct} \]

Step 2: Check Reason.
Consider: \[ x^4+x^3-3x^2-5x-2 \] Factorizing: \[ x^4+x^3-3x^2-5x-2=(x-2)(x+1)^3 \] So: \[ x=-1 \] is a root repeated \(3\) times. \[ R \text{ is correct} \]

Step 3: Check explanation.
Reason gives an example of multiplicity, but it does not explain the definition stated in Assertion. \[ R \text{ is not the correct explanation of } A \] \[ \therefore \text{Correct Answer is (B)} \]