dentify the following given

Question

f
(
x
)
=
(
x
+
5
)
2
(
2
x
−
3
)
5
f(x)=(x+5)
2
(2x−3)
5




Zeros & Multiplicity

For each zero state number of its multiplicity.

​
x
=
−
5
x=−5 multiplicity is


​
x
=
3
2
x=
2
3
​
multiplicity is




Zeros & Effect

For each zero state its effect.

​
x
=
−
5
x=−5 effect is


​
x
=
3
2
x=
2
3
​
effect is


​

Answer 1

To identify the zeros and their multiplicities for the function \( f(x) = (x + 5)^2 (2x - 3)^5 \), we can follow these steps:

Zeros and Their Multiplicities

1. Finding Zeros:

   • For the first factor \( (x + 5)^2 = 0 \): \[ x + 5 = 0 \implies x = -5 \] The multiplicity of the zero at \( x = -5 \) is \( 2 \) (since it is squared).

   • For the second factor \( (2x - 3)^5 = 0 \): \[ 2x - 3 = 0 \implies 2x = 3 \implies x = \frac{3}{2} \] The multiplicity of the zero at \( x = \frac{3}{2} \) is \( 5 \) (since it is raised to the fifth power).

Summary of Zeros and Their Multiplicities:

• \( x = -5 \)

  • Multiplicity: \( 2 \)

• \( x = \frac{3}{2} \)

  • Multiplicity: \( 5 \)

Effect of Each Zero

The effect of a zero can be understood in terms of how it influences the graph of the function at that point:

1. For the zero \( x = -5 \):

   • Since the multiplicity is even (2), the graph touches the x-axis but does not cross it. Thus:
     • Effect: The graph touches the x-axis at \( x = -5 \).

2. For the zero \( x = \frac{3}{2} \):

   • Since the multiplicity is odd (5), the graph crosses the x-axis at this point. Thus:
     • Effect: The graph crosses the x-axis at \( x = \frac{3}{2} \).

Final Summary:

• Zeros & Multiplicities:

  • \( x = -5 \) — Multiplicity: 2
  • \( x = \frac{3}{2} \) — Multiplicity: 5

• Zeros & Effects:

  • \( x = -5 \) — Effect: Touches the x-axis.
  • \( x = \frac{3}{2} \) — Effect: Crosses the x-axis.