Question

If \(\alpha\) is a root of the equation \[ x^4-4x^3+16x-16=0 \] of multiplicity \(m\), then \[ m^2-4m+2= \]

• A. \(-1\)
• B. \(1\)
• C. \(0\)
• D. \(2\) \bigskip

Hint:

To determine the multiplicity of a root, completely factorize the polynomial. The exponent of the factor \((x-\alpha)\) gives the multiplicity of the root \(\alpha\).

Answer 1

The Correct Option is A

Concept: Determine the multiplicity by expressing the polynomial as a product of irreducible factors.

Step 1: Factorize \(x^4-4x^3+16x-16=(x-4)(x^3+4)=(x-4)(x+2)(x^2-2x+4)\).

Step 2: Each factor appears only once, so every root has multiplicity \(m=1\).

Step 3: Therefore, \(m^2-4m+2=1-4+2=-1\). Hence, \(\boxed{-1}\).