Identify the multiplicities of the linear factors of h(x)=(x+4)(x−1)2(x+3)5

ℎ
(
𝑥
)
=
(
𝑥
+
4
)
(
𝑥
−
1
)
2
(
𝑥
+
3
)
5
.(1 point)
Responses

The multiplicity of (x+4)
(
𝑥
+
4
)
is 4, the multiplicity of (x−1)
(
𝑥
−
1
)
is −1
−
1
, and the multiplicity of (x+3)
(
𝑥
+
3
)
is 3.
The multiplicity of left parenthesis x plus 4 right parenthesis is 4, the multiplicity of left parenthesis x minus 1 right parenthesis is negative 1 , and the multiplicity of left parenthesis x plus 3 right parenthesis is 3.

The multiplicity of (x+4)
(
𝑥
+
4
)
is −4
−
4
, the multiplicity of (x−1)
(
𝑥
−
1
)
is 1, and the multiplicity of (x+3)
(
𝑥
+
3
)
is −3
−
3
.
The multiplicity of left parenthesis x plus 4 right parenthesis is negative 4 , the multiplicity of left parenthesis x minus 1 right parenthesis is 1, and the multiplicity of left parenthesis x plus 3 right parenthesis is negative 3 .

The multiplicity of (x+4)
(
𝑥
+
4
)
is 1, the multiplicity of (x−1)
(
𝑥
−
1
)
is 2, and the multiplicity of (x+3)
(
𝑥
+
3
)
is 5.
The multiplicity of left parenthesis x plus 4 right parenthesis is 1, the multiplicity of left parenthesis x minus 1 right parenthesis is 2, and the multiplicity of left parenthesis x plus 3 right parenthesis is 5.

The multiplicity of (x+4)
(
𝑥
+
4
)
is 0, the multiplicity of (x−1)
(
𝑥
−
1
)
is 2, and the multiplicity of (x+3)
(
𝑥
+
3
)
is 5.

1 answer

To identify the multiplicities of the linear factors in the polynomial \( h(x) = (x + 4)(x - 1)^2(x + 3)^5 \), we analyze each factor:

The factor \( (x + 4) \) appears once, so its multiplicity is 1.
The factor \( (x - 1) \) appears squared, so its multiplicity is 2.
The factor \( (x + 3) \) appears raised to the fifth power, so its multiplicity is 5.

Putting it all together:

The multiplicity of \( (x + 4) \) is 1.
The multiplicity of \( (x - 1) \) is 2.
The multiplicity of \( (x + 3) \) is 5.

Therefore, the correct response is:

The multiplicity of \( (x + 4) \) is 1, the multiplicity of \( (x - 1) \) is 2, and the multiplicity of \( (x + 3) \) is 5.