Question
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Part 1
Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value.
nequals3;
negative 2 and 4 plus 5 i are zeros;
f left parenthesis 2 right parenthesis equals 116
Question content area bottom
Part 1
f left parenthesis x right parenthesisequals
enter your response here
(Type an expression using x as the variable. Simplify your answer.)
Part 1
Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value.
nequals3;
negative 2 and 4 plus 5 i are zeros;
f left parenthesis 2 right parenthesis equals 116
Question content area bottom
Part 1
f left parenthesis x right parenthesisequals
enter your response here
(Type an expression using x as the variable. Simplify your answer.)
Answers
GPT-4o mini
To find a polynomial function of degree 3 with the given zeros, we first need to note that the zeros of a polynomial with real coefficients must come in conjugate pairs. You are given the zeros \( -2 \) and \( 4 + 5i \), so the other zero must be \( 4 - 5i \).
The polynomial can be expressed using its factors based on the zeros:
- The zero \( -2 \) corresponds to the factor \( (x + 2) \).
- The zero \( 4 + 5i \) corresponds to the factor \( (x - (4 + 5i)) = (x - 4 - 5i) \).
- The zero \( 4 - 5i \) corresponds to the factor \( (x - (4 - 5i)) = (x - 4 + 5i) \).
Thus, we can write the polynomial \( f(x) \) as:
\[
f(x) = (x + 2) \cdot (x - (4 + 5i)) \cdot (x - (4 - 5i))
\]
Now, let's simplify the factors involving the complex zeros:
1. Multiply the two complex conjugate factors:
\[
(x - (4 + 5i))(x - (4 - 5i)) = ((x - 4) + 5i)((x - 4) - 5i) = (x - 4)^2 - (5i)^2 = (x - 4)^2 + 25
\]
So we can further expand \( (x - 4)^2 + 25 \):
\[
(x - 4)^2 = x^2 - 8x + 16
\]
Thus,
\[
(x - 4)^2 + 25 = x^2 - 8x + 16 + 25 = x^2 - 8x + 41
\]
Now, our polynomial becomes:
\[
f(x) = (x + 2)(x^2 - 8x + 41)
\]
Next, we will expand this product:
\[
f(x) = (x + 2)(x^2 - 8x + 41) = x^3 - 8x^2 + 41x + 2x^2 - 16x + 82
\]
Combine like terms:
\[
f(x) = x^3 - 6x^2 + 25x + 82
\]
Next, we need to ensure that \( f(2) = 116 \). Let's calculate \( f(2) \):
\[
f(2) = (2)^3 - 6(2)^2 + 25(2) + 82
\]
\[
= 8 - 24 + 50 + 82
\]
\[
= 8 - 24 + 132 = 116
\]
This condition is satisfied.
Therefore, the polynomial function is:
\[
\boxed{f(x) = x^3 - 6x^2 + 25x + 82}
\]
The polynomial can be expressed using its factors based on the zeros:
- The zero \( -2 \) corresponds to the factor \( (x + 2) \).
- The zero \( 4 + 5i \) corresponds to the factor \( (x - (4 + 5i)) = (x - 4 - 5i) \).
- The zero \( 4 - 5i \) corresponds to the factor \( (x - (4 - 5i)) = (x - 4 + 5i) \).
Thus, we can write the polynomial \( f(x) \) as:
\[
f(x) = (x + 2) \cdot (x - (4 + 5i)) \cdot (x - (4 - 5i))
\]
Now, let's simplify the factors involving the complex zeros:
1. Multiply the two complex conjugate factors:
\[
(x - (4 + 5i))(x - (4 - 5i)) = ((x - 4) + 5i)((x - 4) - 5i) = (x - 4)^2 - (5i)^2 = (x - 4)^2 + 25
\]
So we can further expand \( (x - 4)^2 + 25 \):
\[
(x - 4)^2 = x^2 - 8x + 16
\]
Thus,
\[
(x - 4)^2 + 25 = x^2 - 8x + 16 + 25 = x^2 - 8x + 41
\]
Now, our polynomial becomes:
\[
f(x) = (x + 2)(x^2 - 8x + 41)
\]
Next, we will expand this product:
\[
f(x) = (x + 2)(x^2 - 8x + 41) = x^3 - 8x^2 + 41x + 2x^2 - 16x + 82
\]
Combine like terms:
\[
f(x) = x^3 - 6x^2 + 25x + 82
\]
Next, we need to ensure that \( f(2) = 116 \). Let's calculate \( f(2) \):
\[
f(2) = (2)^3 - 6(2)^2 + 25(2) + 82
\]
\[
= 8 - 24 + 50 + 82
\]
\[
= 8 - 24 + 132 = 116
\]
This condition is satisfied.
Therefore, the polynomial function is:
\[
\boxed{f(x) = x^3 - 6x^2 + 25x + 82}
\]