Final answer:
To find all the zeros of the polynomial f(x) = x³ - 7x² + 17x - 15 when x - 3 is a factor, we can use synthetic division. After performing the synthetic division, we get a quadratic equation, which can be solved to find the remaining zeros of the polynomial.
Step-by-step explanation:
To determine the zeros of the polynomial f(x) = x³ - 7x² + 17x - 15 when x - 3 is a factor, we can use synthetic division. Set up the synthetic division and divide the polynomial by x - 3. The resulting quotient will be a quadratic equation. We can then solve this quadratic equation to find the remaining zeros of the polynomial.
Step 1:
Set up the synthetic division:
3 | 1 -7 17 -15
Step 2:
Perform the synthetic division:
1 -4 5
Step 3:
Write the resulting quadratic equation: x² - 4x + 5 = 0
Step 4:
Solve the quadratic equation: x = 2 ± i
Therefore, the zeros of the polynomial f(x) = x³ - 7x² + 17x - 15, given that x - 3 is a factor, are 3, 2 + i, and 2 - i.