Answer :
To find an nth-degree polynomial function with real coefficients, given that the polynomial has the degree [tex]\( n = 4 \)[/tex] and the zeros [tex]\( i \)[/tex] and [tex]\( 5i \)[/tex], we can proceed as follows:
1. Identify the Zeros and Their Conjugates:
- If [tex]\( i \)[/tex] and [tex]\( 5i \)[/tex] are zeros and the coefficients are real, their complex conjugates [tex]\( -i \)[/tex] and [tex]\( -5i \)[/tex] must also be zeros.
2. Write the Polynomial with the Given Zeros:
- The zeros [tex]\((i, -i)\)[/tex] can form the factor [tex]\((x - i)(x + i)\)[/tex].
- The zeros [tex]\((5i, -5i)\)[/tex] can form the factor [tex]\((x - 5i)(x + 5i)\)[/tex].
3. Simplify the Polynomial Factors:
- The product [tex]\((x - i)(x + i)\)[/tex] simplifies using the difference of squares: [tex]\(x^2 + 1\)[/tex].
- The product [tex]\((x - 5i)(x + 5i)\)[/tex] simplifies using the difference of squares: [tex]\(x^2 + 25\)[/tex].
4. Construct the Polynomial:
- Multiply the simplified factors: [tex]\((x^2 + 1)(x^2 + 25)\)[/tex].
5. Expand the Polynomial:
[tex]\[
(x^2 + 1)(x^2 + 25) = x^4 + 25x^2 + x^2 + 25 = x^4 + 26x^2 + 25
\][/tex]
So, the polynomial is [tex]\( f(x) = x^4 + 26x^2 + 25 \)[/tex].
6. Scale the Polynomial to Satisfy the Given Condition [tex]\( f(-2) = 145 \)[/tex]:
- Substitute [tex]\( x = -2 \)[/tex] into [tex]\( f(x) \)[/tex] to calculate the current value:
[tex]\[
f(-2) = (-2)^4 + 26(-2)^2 + 25 = 16 + 26 \times 4 + 25 = 16 + 104 + 25 = 145
\][/tex]
- Since we already have the value [tex]\( f(-2) = 145 \)[/tex] matching the condition, we confirm that no scaling is needed.
Therefore, the polynomial function that satisfies the given conditions is:
[tex]\[
f(x) = x^4 + 26x^2 + 25
\][/tex]
1. Identify the Zeros and Their Conjugates:
- If [tex]\( i \)[/tex] and [tex]\( 5i \)[/tex] are zeros and the coefficients are real, their complex conjugates [tex]\( -i \)[/tex] and [tex]\( -5i \)[/tex] must also be zeros.
2. Write the Polynomial with the Given Zeros:
- The zeros [tex]\((i, -i)\)[/tex] can form the factor [tex]\((x - i)(x + i)\)[/tex].
- The zeros [tex]\((5i, -5i)\)[/tex] can form the factor [tex]\((x - 5i)(x + 5i)\)[/tex].
3. Simplify the Polynomial Factors:
- The product [tex]\((x - i)(x + i)\)[/tex] simplifies using the difference of squares: [tex]\(x^2 + 1\)[/tex].
- The product [tex]\((x - 5i)(x + 5i)\)[/tex] simplifies using the difference of squares: [tex]\(x^2 + 25\)[/tex].
4. Construct the Polynomial:
- Multiply the simplified factors: [tex]\((x^2 + 1)(x^2 + 25)\)[/tex].
5. Expand the Polynomial:
[tex]\[
(x^2 + 1)(x^2 + 25) = x^4 + 25x^2 + x^2 + 25 = x^4 + 26x^2 + 25
\][/tex]
So, the polynomial is [tex]\( f(x) = x^4 + 26x^2 + 25 \)[/tex].
6. Scale the Polynomial to Satisfy the Given Condition [tex]\( f(-2) = 145 \)[/tex]:
- Substitute [tex]\( x = -2 \)[/tex] into [tex]\( f(x) \)[/tex] to calculate the current value:
[tex]\[
f(-2) = (-2)^4 + 26(-2)^2 + 25 = 16 + 26 \times 4 + 25 = 16 + 104 + 25 = 145
\][/tex]
- Since we already have the value [tex]\( f(-2) = 145 \)[/tex] matching the condition, we confirm that no scaling is needed.
Therefore, the polynomial function that satisfies the given conditions is:
[tex]\[
f(x) = x^4 + 26x^2 + 25
\][/tex]
