Question
Use the graph of the polynomial function f(x) (a) to solve f(x)equals0, and (b) find the factorization of f(x).
f left parenthesis x right parenthesis equals x cubed minus 4 x squared minus 4 x plus 16
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Part 1
(a) xequals
enter your response here
(Use a comma to separate answers as needed. Type an integer or a simplified fraction.)
f left parenthesis x right parenthesis equals x cubed minus 4 x squared minus 4 x plus 16
Question content area bottom left
Part 1
(a) xequals
enter your response here
(Use a comma to separate answers as needed. Type an integer or a simplified fraction.)
Answers
GPT-4o mini
To solve \( f(x) = 0 \) for the polynomial function \( f(x) = x^3 - 4x^2 - 4x + 16 \), you can use techniques such as synthetic division, factoring, or the Rational Root Theorem in the absence of a graph.
### Step 1: Finding the roots
1. Identify possible rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term (16) divided by the factors of the leading coefficient (1). The factors of 16 are \( \pm 1, \pm 2, \pm 4, \pm 8, \pm 16 \).
2. Test these possible roots by evaluating \( f(x) \):
- \( f(1) = 1^3 - 4(1^2) - 4(1) + 16 = 1 - 4 - 4 + 16 = 9 \) (not a root)
- \( f(-1) = (-1)^3 - 4(-1)^2 - 4(-1) + 16 = -1 - 4 + 4 + 16 = 15 \) (not a root)
- \( f(2) = 2^3 - 4(2^2) - 4(2) + 16 = 8 - 16 - 8 + 16 = 0 \) (is a root)
- \( f(-2) = (-2)^3 - 4(-2)^2 - 4(-2) + 16 = -8 - 16 + 8 + 16 = 0 \) (is a root)
3. Since \( x = 2 \) and \( x = -2 \) are roots, we can factor \( f(x) \) using \( (x - 2) \) and \( (x + 2) \).
### Step 2: Polynomial long division or synthetic division
To fully factor the polynomial, first divide \( f(x) \) by \( (x - 2)(x + 2) = (x^2 - 4) \).
Perform synthetic or long division:
\[
x^3 - 4x^2 - 4x + 16 \div (x^2 - 4)
\]
The result of this is \( x - 4 \).
### Step 3: Complete Factorization
Thus we have:
\[
f(x) = (x - 2)(x + 2)(x - 4)
\]
### Final Answer
#### (a) The solutions to \( f(x) = 0 \) are:
\( x = 2, -2, 4 \)
#### (b) The factorization of \( f(x) \) is:
\[
f(x) = (x - 2)(x + 2)(x - 4)
\]
Please fill in the values provided above in your answer boxes.
### Step 1: Finding the roots
1. Identify possible rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term (16) divided by the factors of the leading coefficient (1). The factors of 16 are \( \pm 1, \pm 2, \pm 4, \pm 8, \pm 16 \).
2. Test these possible roots by evaluating \( f(x) \):
- \( f(1) = 1^3 - 4(1^2) - 4(1) + 16 = 1 - 4 - 4 + 16 = 9 \) (not a root)
- \( f(-1) = (-1)^3 - 4(-1)^2 - 4(-1) + 16 = -1 - 4 + 4 + 16 = 15 \) (not a root)
- \( f(2) = 2^3 - 4(2^2) - 4(2) + 16 = 8 - 16 - 8 + 16 = 0 \) (is a root)
- \( f(-2) = (-2)^3 - 4(-2)^2 - 4(-2) + 16 = -8 - 16 + 8 + 16 = 0 \) (is a root)
3. Since \( x = 2 \) and \( x = -2 \) are roots, we can factor \( f(x) \) using \( (x - 2) \) and \( (x + 2) \).
### Step 2: Polynomial long division or synthetic division
To fully factor the polynomial, first divide \( f(x) \) by \( (x - 2)(x + 2) = (x^2 - 4) \).
Perform synthetic or long division:
\[
x^3 - 4x^2 - 4x + 16 \div (x^2 - 4)
\]
The result of this is \( x - 4 \).
### Step 3: Complete Factorization
Thus we have:
\[
f(x) = (x - 2)(x + 2)(x - 4)
\]
### Final Answer
#### (a) The solutions to \( f(x) = 0 \) are:
\( x = 2, -2, 4 \)
#### (b) The factorization of \( f(x) \) is:
\[
f(x) = (x - 2)(x + 2)(x - 4)
\]
Please fill in the values provided above in your answer boxes.