Question

find the root(s) of (f(x) = (x - 6)^2(x + 2)^2).
-6 with multiplicity 1
-6 with multiplicity 2
6 with multiplicity 1
6 with multiplicity 2
-2 with multiplicity 1
-2 with multiplicity 2
2 with multiplicity 1
2 with multiplicity 2

find the root(s) of (f(x) = (x + 5)^3(x - 9)^2(x + 1)).
-5 with multiplicity 3
5 with multiplicity 3
-9 with multiplicity 2
9 with multiplicity 2
-1 with multiplicity 0
-1 with multiplicity 1
1 with multiplicity 0
1 with multiplicity 1

Explanation:

⚡ Using what you learned: Zeros and Multiplicity

Step 1: Analyze the factor \( (x + 5)^3 \)

Set the factor equal to zero to find the root:
\[ x + 5 = 0 \implies x = -5 \]

The exponent of this factor is \( 3 \), which represents its multiplicity.

• Root: \( -5 \) with multiplicity \( 3 \)

Step 2: Analyze the factor \( (x - 9)^2 \)

Set the factor equal to zero to find the root:
\[ x - 9 = 0 \implies x = 9 \]

The exponent of this factor is \( 2 \), which represents its multiplicity.

• Root: \( 9 \) with multiplicity \( 2 \)

Step 3: Analyze the factor \( (x + 1) \)

Set the factor equal to zero to find the root:
\[ x + 1 = 0 \implies x = -1 \]

The exponent of this factor is understood to be \( 1 \), which represents its multiplicity.

• Root: \( -1 \) with multiplicity \( 1 \)

Answer:

The correct options are:

• -5 with multiplicity 3
• 9 with multiplicity 2
• -1 with multiplicity 1