For the polynomial function below: (a) List each real zero and its multiplicity. (b) Deter
whether the graph crosses or touches the -axis at each -intercept. © Determine the
end behavior.

(a) Find any real zeros of .
The real zero(s) of is/are
The multiplicity of the smaller zero is
(Type a whole number.)
The multiplicity of the larger zero is .
(Type a whole number.)
(b) The graph the -axis at the smaller -intercept.
The graph the -axis at the larger -intercept.

Ask by Savage Rogers. in the United States

Dec 12,2024

Question

For the polynomial function below: (a) List each real zero and its multiplicity. (b) Deter
whether the graph crosses or touches the -axis at each -intercept. © Determine the
end behavior.

(a) Find any real zeros of .
The real zero(s) of is/are
The multiplicity of the smaller zero is
(Type a whole number.)
The multiplicity of the larger zero is .
(Type a whole number.)
(b) The graph the -axis at the smaller -intercept.
The graph the -axis at the larger -intercept.

Ask by Savage Rogers. in the United States

Dec 12,2024

UpStudy AI · Accepted Answer

To analyze the polynomial function $$ f(x) = (x-4)^{2}(x+2)^{3} $$, we will go through each part of the question step by step.

### (a) Find any real zeros of $$ f $$.

1. **Finding the real zeros:**
   - The real zeros occur when $$ f(x) = 0 $$.
   - Setting each factor to zero:
     - $$ (x - 4)^{2} = 0 $$ gives $$ x = 4 $$.
     - $$ (x + 2)^{3} = 0 $$ gives $$ x = -2 $$.

2. **Listing the real zeros:**
   - The real zeros of $$ f $$ are $$ x = 4 $$ and $$ x = -2 $$.

3. **Finding the multiplicities:**
   - The multiplicity of the zero $$ x = 4 $$ is $$ 2 $$ (since it is raised to the power of 2).
   - The multiplicity of the zero $$ x = -2 $$ is $$ 3 $$ (since it is raised to the power of 3).

Thus, we have:
- The real zero(s) of $$ f $$ is/are $$ -2, 4 $$.
- The multiplicity of the smaller zero (which is $$ -2 $$) is $$ 3 $$.
- The multiplicity of the larger zero (which is $$ 4 $$) is $$ 2 $$.

### (b) Determine whether the graph crosses or touches the $$ x $$-axis at each $$ x $$-intercept.

1. **Behavior at $$ x = -2 $$:**
   - Since the multiplicity of the zero $$ -2 $$ is $$ 3 $$ (an odd number), the graph **crosses** the $$ x $$-axis at this intercept.

2. **Behavior at $$ x = 4 $$:**
   - Since the multiplicity of the zero $$ 4 $$ is $$ 2 $$ (an even number), the graph **touches** the $$ x $$-axis at this intercept.

Thus, we have:
- The graph **crosses** the $$ x $$-axis at the smaller $$ x $$-intercept $$ -2 $$.
- The graph **touches** the $$ x $$-axis at the larger $$ x $$-intercept $$ 4 $$.

### (c) Determine the end behavior.

To determine the end behavior of the polynomial, we look at the leading term of the polynomial. The leading term can be found by expanding $$ f(x) $$:

1. The degree of $$ f(x) $$ is $$ 2 + 3 = 5 $$ (since the highest power of $$ x $$ in the factors is $$ 2 $$ from $$ (x-4)^{2} $$ and $$ 3 $$ from $$ (x+2)^{3} $$).
2. The leading coefficient is positive (since the leading term of $$ (x-4)^{2} $$ is $$ x^{2} $$ and the leading term of $$ (x+2)^{3} $$ is $$ x^{3} $$, and both are positive).

Thus, the end behavior of $$ f(x) $$ is:
- As $$ x 	o -\infty $$, $$ f(x) 	o -\infty $$.
- As $$ x 	o +\infty $$, $$ f(x) 	o +\infty $$.

### Summary of Answers:
(a) The real zero(s) of $$ f $$ is/are $$ -2, 4 $$. The multiplicity of the smaller zero is $$ 3 $$. The multiplicity of the larger zero is $$ 2 $$.

(b) The graph **crosses** the $$ x $$-axis at the smaller $$ x $$-intercept. The graph **touches** the $$ x $$-axis at the larger $$ x $$-intercept.

(c) The end behavior is:
- As $$ x 	o -\infty $$, $$ f(x) 	o -\infty $$.
- As $$ x 	o +\infty $$, $$ f(x) 	o +\infty $$.
(a) The real zeros of $$ f $$ are $$ -2, 4 $$. The multiplicity of the smaller zero is $$ 3 $$. The multiplicity of the larger zero is $$ 2 $$. (b) The graph crosses the $$ x $$-axis at the smaller $$ x $$-intercept. The graph touches the $$ x $$-axis at the larger $$ x $$-intercept. (c) As $$ x 	o -\infty $$, $$ f(x) 	o -\infty $$. As $$ x 	o +\infty $$, $$ f(x) 	o +\infty $$.

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