Sure, let's find the polynomial function step-by-step given the roots [tex]\( -2 \)[/tex], [tex]\( 4 \)[/tex], [tex]\( 3 + i \)[/tex], and [tex]\( 3 - i \)[/tex], and the condition [tex]\( f(-3) = 15 \)[/tex].
### Step-by-Step Solution:
1. Identify the roots and their implications:
- The given roots are [tex]\( -2 \)[/tex], [tex]\( 4 \)[/tex], [tex]\( 3 + i \)[/tex], and [tex]\( 3 - i \)[/tex]. Since the polynomial must have real coefficients, the complex roots have to appear in conjugate pairs [tex]\( (3 + i) \)[/tex] and [tex]\( (3 - i) \)[/tex].
2. Form the polynomial from the roots:
- A polynomial can be written as the product of its factors based on its roots. Thus, the polynomial [tex]\( f(x) \)[/tex] with these roots can be written as:
[tex]\[
f(x) = k (x + 2)(x - 4)\left( x - (3+i) \right)\left( x - (3-i) \right)
\][/tex]
where [tex]\( k \)[/tex] is a constant multiplier to be determined.
3. Simplify the polynomial:
- First, simplify the factors involving the complex roots:
[tex]\[
(x - (3+i))(x - (3-i)) = \left((x - 3) - i\right)\left((x - 3) + i\right)
\][/tex]
Using the difference of squares:
[tex]\[
(x - 3)^2 - i^2 = (x - 3)^2 + 1 = x^2 - 6x + 9 + 1 = x^2 - 6x + 10
\][/tex]
- Now the polynomial becomes:
[tex]\[
f(x) = k (x + 2)(x - 4)(x^2 - 6x + 10)
\][/tex]
4. Expand the polynomial:
- Multiply out the factors:
[tex]\[
(x + 2)(x - 4) = x^2 - 4x + 2x - 8 = x^2 - 2x - 8
\][/tex]
[tex]\[
f(x) = k (x^2 - 2x - 8)(x^2 - 6x + 10)
\][/tex]
Expanding this product:
[tex]\[
(x^2 - 2x - 8)(x^2 - 6x + 10)
\ = x^4 - 6x^3 + 10x^2 - 2x^3 + 12x^2 - 20x - 8x^2 + 48x - 80
\ = x^4 - 8x^3 + 14x^2 + 28x - 80
\][/tex]
5. Include the constant multiplier [tex]\( k \)[/tex]:
- Now we have:
[tex]\[
f(x) = k \left( x^4 - 8x^3 + 14x^2 + 28x - 80 \right)
\][/tex]
6. Determine [tex]\( k \)[/tex] using the condition [tex]\( f(-3) = 15 \)[/tex]:
- Substitute [tex]\( x = -3 \)[/tex] and [tex]\( f(-3) = 15 \)[/tex]:
[tex]\[
15 = k \left( (-3)^4 - 8(-3)^3 + 14(-3)^2 + 28(-3) - 80 \right)
\][/tex]
Calculate the value inside the brackets:
[tex]\[
15 = k \left( 81 + 216 + 126 - 84 - 80 \right)
= k \left( 81 + 216 + 126 - 84 - 80 \right)
= k (259)
\][/tex]
So,
[tex]\[
k = \frac{15}{259}
\][/tex]
7. Write the final polynomial:
- Substitute the value of [tex]\( k \)[/tex] back into the polynomial:
[tex]\[
f(x) = \frac{15}{259} \left( x^4 - 8x^3 + 14x^2 + 28x - 80 \right)
\][/tex]
Distribute the constant:
[tex]\[
f(x) = \frac{15}{259} x^4 - \frac{120}{259} x^3 + \frac{210}{259} x^2 + \frac{420}{259} x - \frac{1200}{259}
\][/tex]
Therefore, the polynomial function of degree 4 with only real coefficients based on the given roots and condition is:
[tex]\[
f(x) = \frac{15}{259} x^4 - \frac{120}{259} x^3 + \frac{210}{259} x^2 + \frac{420}{259} x - \frac{1200}{259}
\][/tex]
