Use the graph of the polynomial function​ f(x) ​(a) to solve ​f(x)equals​0, and ​(b) find the factorization of​ f(x).

Question

​f(x)equalsnegative 2 x Superscript 4 Baseline plus 6 x cubed plus 10 x squared minus 6 x minus 8
Question content area bottom left
Part 1
​(a) Choose the correct solution for ​f(x)equals0.

A.
xequalsminus1 and xequalsminus4
B.
xequals1 and xequals4
C.
xequals1​, xequalsminus1​, and xequalsminus4
D.
xequalsminus1​, xequals1​, and xequals4
Part 2
​(b) The factorization of the polynomial function​ f(x) is
  
enter your response here.
​(Type your answer in factored​ form.)
.
.
.
Question content area right
Part 1
-2
2
4
6
-100
-80
-60
-40
-20
20
40
60
x
y

A coordinate system has a horizontal x-axis labeled from negative 3 to 6 in increments of 1 and a vertical y-axis labeled from negative 100 to 70 in increments of 10. From left to right, a curve rises in quadrant 3 to a maximum 1 unit to the left of the origin, falls to a minimum in quadrant 4, rises and passes through a point 1 unit to the right of the origin to a maximum in quadrant 1, and falls passing through a point 4 units to the right of the origin.

Answer 1

To solve the problem, we start with the given polynomial function:

\[ f(x) = -2x^4 + 6x^3 + 10x^2 - 6x - 8 \]

Part 1: Solving \( f(x) = 0 \)

To determine where \( f(x) = 0 \), we can examine the roots that correspond to where the graph intersects the x-axis:

Based on your description of the graph:

• It passes through the x-axis at approximately \( x = -1 \)
• It also has a root around \( x = 1 \)
• Finally, there is another root around \( x = 4 \)

Thus, the solutions for \( f(x) = 0 \) appear to be \( x = -1, x = 1, \) and \( x = 4 \).

This aligns with option D. Therefore, the solution is:

Answer for Part 1: D. \( x = -1, x = 1, \) and \( x = 4 \)

Part 2: Factorization of \( f(x) \)

To find the factorization of \( f(x) \), we can express \( f(x) \) as a product of its factors based on the roots we discovered:

Since the roots are \( x = -1, x = 1, \) and \( x = 4 \), the corresponding factors will be:

• For \( x = -1 \): \( (x + 1) \)
• For \( x = 1 \): \( (x - 1) \)
• For \( x = 4 \): \( (x - 4) \)

Considering the multiplicity of the roots and the leading coefficient:

The polynomial can be expressed in factored form. Since the leading coefficient of \( f(x) \) is \( -2 \), the complete factorization incorporating the leading coefficient would be:

\[ f(x) = -2(x + 1)(x - 1)(x - 4) \]

Thus, the factorization of \( f(x) \) is:

Answer for Part 2: \( f(x) = -2(x + 1)(x - 1)(x - 4) \)

In conclusion:

• Part 1: D. \( x = -1, x = 1, x = 4 \)
• Part 2: \( f(x) = -2(x + 1)(x - 1)(x - 4) \)