Question

which zero has multiplicity 2 in p(x) = ((x + 1)^2)(x - 2)(x + 3)(x - 1)? options: x=2, x=1, x=-3, x=-1

Explanation:

Step1: Recall multiplicity definition

The multiplicity of a zero \( x = a \) in a polynomial \( p(x) \) is the exponent of the factor \( (x - a) \) (or \( (a - x) \) rewritten as \( (x - a) \) with a sign change) in the factored form of \( p(x) \).

Step2: Analyze each factor's exponent

Given \( p(x)=(x + 1)^2(x - 2)(x + 3)(x - 1) \):

• For \( x = 2 \): The factor is \( (x - 2) \), exponent is \( 1 \), so multiplicity \( 1 \).
• For \( x = 1 \): The factor is \( (x - 1) \), exponent is \( 1 \), so multiplicity \( 1 \).
• For \( x=- 3 \): The factor is \( (x + 3)=(x-(-3)) \), exponent is \( 1 \), so multiplicity \( 1 \).
• For \( x=-1 \): The factor is \( (x + 1)=(x-(-1)) \), exponent is \( 2 \), so multiplicity \( 2 \).

Answer:

\( x = -1 \) (corresponding to the option with \( x=-1 \))