Question

determine the multiplicity of the roots of the function
$k(x) = x(x + 2)^3(x + 4)^2(x - 5)^4$.

0 has multiplicity

-2 has multiplicity

-4 has multiplicity

5 has multiplicity

Explanation:

Step1: Recall root multiplicity

For a polynomial in factored form \( k(x) = (x - a)^n \), the root \( a \) has multiplicity \( n \).

Step2: Analyze root \( 0 \)

The factor is \( x = (x - 0)^1 \), so multiplicity is \( 1 \).

Step3: Analyze root \( -2 \)

The factor is \( (x + 2)^3 = (x - (-2))^3 \), so multiplicity is \( 3 \).

Step4: Analyze root \( -4 \)

The factor is \( (x + 4)^2 = (x - (-4))^2 \), so multiplicity is \( 2 \).

Step5: Analyze root \( 5 \)

The factor is \( (x - 5)^4 \), so multiplicity is \( 4 \).

Answer:

0 has multiplicity 1
-2 has multiplicity 3
-4 has multiplicity 2
5 has multiplicity 4