Question

consider the following.\\( f(x) = x^4 - x^3 - 12x^2 \\)\\( \text{(a) find all real zeros of the polynomial function. (enter your answers as a comma-separated list. if there is no solution, enter no solution.)} \\)\\( x = \\)\\( \text{(b) determine the multiplicity of each zero.} \\)\\( \text{smallest } x\text{-value} \\)\\( \text{largest } x\text{-value} \\)\\( \text{(c) determine the maximum possible number of turning points of the graph of the function.} \\)\\( \text{(d) use a graphing utility to graph the function and verify your answers.} \\)

Explanation:

Part (a)

Step 1: Factor the polynomial

To find the real zeros of \( f(x) = x^4 - x^3 - 12x^2 \), we start by factoring out the greatest common factor, which is \( x^2 \).
\[
f(x) = x^2(x^2 - x - 12)
\]

Step 2: Factor the quadratic

Next, we factor the quadratic \( x^2 - x - 12 \). We need two numbers that multiply to \( -12 \) and add to \( -1 \). These numbers are \( -4 \) and \( 3 \). So,
\[
x^2 - x - 12 = (x - 4)(x + 3)
\]

Step 3: Find the zeros

Now we have the factored form of the polynomial:
\[
f(x) = x^2(x - 4)(x + 3)
\]
To find the real zeros, we set \( f(x) = 0 \) and solve for \( x \):

• \( x^2 = 0 \) gives \( x = 0 \) (with multiplicity 2)
• \( x - 4 = 0 \) gives \( x = 4 \)
• \( x + 3 = 0 \) gives \( x = -3 \)

Step 1: Recall the rule for turning points

The maximum number of turning points of a polynomial function of degree \( n \) is \( n - 1 \).

Step 2: Determine the degree of the polynomial

The polynomial \( f(x) = x^4 - x^3 - 12x^2 \) is a fourth-degree polynomial (since the highest power of \( x \) is 4).

Step 3: Calculate the maximum number of turning points

Using the rule, for a degree 4 polynomial, the maximum number of turning points is \( 4 - 1 = 3 \).

Answer:

\( -3, 0, 4 \)

Part (b)

For the zero \( x = -3 \):

• The factor is \( (x + 3) \), which has an exponent of 1 (odd multiplicity).

For the zero \( x = 0 \):

• The factor is \( x^2 \), which has an exponent of 2 (even multiplicity).

For the zero \( x = 4 \):

• The factor is \( (x - 4) \), which has an exponent of 1 (odd multiplicity).

Smallest \( x \)-value: \( x = -3 \) (odd multiplicity)
Largest \( x \)-value: \( x = 4 \) (odd multiplicity)

Part (c)