Answer :
To find an nth degree polynomial function with real coefficients that satisfies the given conditions, follow these steps:
1. Identify the Given Zeros:
The zeros given are [tex]\(3\)[/tex], [tex]\(\frac{1}{3}\)[/tex], and [tex]\(1+2i\)[/tex].
Since polynomial functions with real coefficients must have complex conjugate pairs as zeros, [tex]\(1-2i\)[/tex] must also be a zero.
2. Form the Polynomial:
The polynomial can be constructed by using these zeros. The polynomial [tex]\(f(x)\)[/tex] can be expressed as:
[tex]\[
f(x) = (x - 3)(x - \frac{1}{3})(x - (1 + 2i))(x - (1 - 2i))
\][/tex]
3. Simplify the Polynomial:
We'll first handle the complex conjugates:
[tex]\[
(x - (1 + 2i))(x - (1 - 2i)) = (x - 1 - 2i)(x - 1 + 2i)
\][/tex]
Using the identity [tex]\((a - b)(a + b) = a^2 - b^2\)[/tex]:
[tex]\[
= ((x - 1)^2 - (2i)^2) = (x - 1)^2 + 4
\][/tex]
Expanding [tex]\((x - 1)^2\)[/tex]:
[tex]\[
= (x^2 - 2x + 1) + 4 = x^2 - 2x + 5
\][/tex]
4. Combine All Parts:
Now, multiply all the factors together:
[tex]\[
f(x) = (x - 3)(x - \frac{1}{3})(x^2 - 2x + 5)
\][/tex]
Multiply the first two linear terms:
[tex]\[
(x - 3)(x - \frac{1}{3}) = x^2 - x \cdot \frac{1}{3} - 3x + 1 = x^2 - \frac{10}{3}x + 1
\][/tex]
5. Form the Polynomial:
[tex]\[
f(x) = (x^2 - \frac{10}{3}x + 1)(x^2 - 2x + 5)
\][/tex]
6. Find Scaling Factor:
Expand this polynomial and find the value for [tex]\(f(1)\)[/tex]. Calculate the scaling factor needed to adjust the polynomial so [tex]\(f(1) = 48\)[/tex]. Multiply the polynomial by this factor.
7. Final Polynomial Expression:
After the scaling, the correct polynomial that satisfies the given conditions is:
[tex]\[
f(x) = -9x^4 + 48x^3 - 114x^2 + 168x - 45
\][/tex]
Thus, the polynomial function that satisfies all given conditions is:
[tex]\[
\boxed{-9x^4 + 48x^3 - 114x^2 + 168x - 45}
\][/tex]
Note: Be sure to double-check this solution by plugging in [tex]\(x = 1\)[/tex] to ensure it equals 48, further verifying the correctness of the polynomial.
1. Identify the Given Zeros:
The zeros given are [tex]\(3\)[/tex], [tex]\(\frac{1}{3}\)[/tex], and [tex]\(1+2i\)[/tex].
Since polynomial functions with real coefficients must have complex conjugate pairs as zeros, [tex]\(1-2i\)[/tex] must also be a zero.
2. Form the Polynomial:
The polynomial can be constructed by using these zeros. The polynomial [tex]\(f(x)\)[/tex] can be expressed as:
[tex]\[
f(x) = (x - 3)(x - \frac{1}{3})(x - (1 + 2i))(x - (1 - 2i))
\][/tex]
3. Simplify the Polynomial:
We'll first handle the complex conjugates:
[tex]\[
(x - (1 + 2i))(x - (1 - 2i)) = (x - 1 - 2i)(x - 1 + 2i)
\][/tex]
Using the identity [tex]\((a - b)(a + b) = a^2 - b^2\)[/tex]:
[tex]\[
= ((x - 1)^2 - (2i)^2) = (x - 1)^2 + 4
\][/tex]
Expanding [tex]\((x - 1)^2\)[/tex]:
[tex]\[
= (x^2 - 2x + 1) + 4 = x^2 - 2x + 5
\][/tex]
4. Combine All Parts:
Now, multiply all the factors together:
[tex]\[
f(x) = (x - 3)(x - \frac{1}{3})(x^2 - 2x + 5)
\][/tex]
Multiply the first two linear terms:
[tex]\[
(x - 3)(x - \frac{1}{3}) = x^2 - x \cdot \frac{1}{3} - 3x + 1 = x^2 - \frac{10}{3}x + 1
\][/tex]
5. Form the Polynomial:
[tex]\[
f(x) = (x^2 - \frac{10}{3}x + 1)(x^2 - 2x + 5)
\][/tex]
6. Find Scaling Factor:
Expand this polynomial and find the value for [tex]\(f(1)\)[/tex]. Calculate the scaling factor needed to adjust the polynomial so [tex]\(f(1) = 48\)[/tex]. Multiply the polynomial by this factor.
7. Final Polynomial Expression:
After the scaling, the correct polynomial that satisfies the given conditions is:
[tex]\[
f(x) = -9x^4 + 48x^3 - 114x^2 + 168x - 45
\][/tex]
Thus, the polynomial function that satisfies all given conditions is:
[tex]\[
\boxed{-9x^4 + 48x^3 - 114x^2 + 168x - 45}
\][/tex]
Note: Be sure to double-check this solution by plugging in [tex]\(x = 1\)[/tex] to ensure it equals 48, further verifying the correctness of the polynomial.
