To write a polynomial [tex]\( f(x) \)[/tex] of degree 3 with integer coefficients that has zeros [tex]\(-5i\)[/tex] and [tex]\(\frac{8}{5}\)[/tex], let's follow a step-by-step process.
### Step 1: Identify the Zeros
Given that the polynomial has zeros at [tex]\(-5i\)[/tex] and [tex]\(\frac{8}{5}\)[/tex], and knowing that coefficients must be integers, any complex zero must have its conjugate also as a zero (to ensure we end up with a polynomial with real coefficients).
Thus, the zeros are:
1. [tex]\(-5i\)[/tex]
2. [tex]\(5i\)[/tex] (the conjugate of [tex]\(-5i\)[/tex])
3. [tex]\(\frac{8}{5}\)[/tex]
### Step 2: Form Factors from Zeros
From these zeros, we form linear factors:
1. [tex]\((x + 5i)\)[/tex]
2. [tex]\((x - 5i)\)[/tex]
3. [tex]\(\left(x - \frac{8}{5}\right)\)[/tex]
### Step 3: Combine Complex Conjugate Factors
The product of the complex conjugate factors [tex]\((x + 5i)\)[/tex] and [tex]\((x - 5i)\)[/tex] yields a quadratic polynomial with real coefficients:
[tex]\[
(x + 5i)(x - 5i) = x^2 - (5i)^2 = x^2 - (25(-1)) = x^2 + 25
\][/tex]
### Step 4: Include the Rational Zero
Next, we incorporate the factor from the rational zero [tex]\(\frac{8}{5}\)[/tex]:
[tex]\[
\left(x - \frac{8}{5}\right)
\][/tex]
But, to make all coefficients integers, we multiply this factor by 5 (the denominator):
[tex]\[
5 \left(x - \frac{8}{5}\right) = 5x - 8
\][/tex]
### Step 5: Form the Polynomial
Combine the quadratic polynomial [tex]\(x^2 + 25\)[/tex] and the linear term [tex]\(5x - 8\)[/tex]:
[tex]\[
f(x) = (x^2 + 25)(5x - 8)
\][/tex]
### Step 6: Expand the Polynomial
Finally, expand this product to obtain the polynomial in standard form:
[tex]\[
f(x) = (x^2 + 25)(5x - 8)
\][/tex]
Expanding each term:
1. [tex]\(x^2 \cdot 5x = 5x^3\)[/tex]
2. [tex]\(x^2 \cdot (-8) = -8x^2\)[/tex]
3. [tex]\(25 \cdot 5x = 125x\)[/tex]
4. [tex]\(25 \cdot (-8) = -200\)[/tex]
Combining these terms:
[tex]\[
f(x) = 5x^3 - 8x^2 + 125x - 200
\][/tex]
Thus, the polynomial [tex]\(f(x)\)[/tex] which satisfies the given conditions is:
[tex]\[
\boxed{5x^3 - 8x^2 + 125x - 200}
\][/tex]
