Question

you must submit thorough, handwritten work. you may only use a writing utensil, paper, and a calculator that does not have a computer algebra system. find all the zeroes of the polynomial function $f(x)=x^3 - 5x^2 + 6x - 30$. if you use synthetic division, show all three lines of numbers.

Explanation:

Step1: Try to find a real root by Rational Root Theorem

The possible rational roots are factors of 30 over factors of 1, so possible roots: ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30.
Test \(x = 5\): \(f(5)=5^{3}-5\times5^{2}+6\times5 - 30=125 - 125+30 - 30 = 0\). So \(x = 5\) is a root.

Step2: Perform synthetic division with root 5

Set up synthetic division:
Coefficients: 1 (for \(x^{3}\)), -5 (for \(x^{2}\)), 6 (for \(x\)), -30 (constant)
Bring down the 1.
Multiply 1 by 5: 5. Add to -5: 0.
Multiply 0 by 5: 0. Add to 6: 6.
Multiply 6 by 5: 30. Add to -30: 0.
So the quotient polynomial is \(x^{2}+0x + 6=x^{2}+6\).

Step3: Solve \(x^{2}+6 = 0\)

\(x^{2}=-6\), so \(x=\pm\sqrt{-6}=\pm i\sqrt{6}\).

Answer:

The zeroes of \(f(x)\) are \(5\), \(i\sqrt{6}\), and \(-i\sqrt{6}\).