Question

find all zeros of $f(x) = x^3 - 7x^2 + 9x + 5$. enter the zeros separated by commas. enter exact value, not decimal approximations.

Explanation:

Step1: Test rational root candidates

By Rational Root Theorem, possible roots are $\pm1, \pm5$. Test $x=1$:
$1^3 -7(1)^2 +9(1)+5 = 1-7+9+5=8
eq0$
Test $x=5$:
$5^3 -7(5)^2 +9(5)+5 = 125-175+45+5=0$
So $x=5$ is a root, $(x-5)$ is a factor.

Step2: Polynomial division

Divide $x^3-7x^2+9x+5$ by $(x-5)$:
Using synthetic division:

5 | 1  -7   9   5
        5  -10  -5
      1  -2  -1   0

Quotient: $x^2-2x-1$

Step3: Solve quadratic equation

Solve $x^2-2x-1=0$ using quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=1, b=-2, c=-1$:
$x=\frac{2\pm\sqrt{(-2)^2-4(1)(-1)}}{2(1)}=\frac{2\pm\sqrt{4+4}}{2}=\frac{2\pm\sqrt{8}}{2}=\frac{2\pm2\sqrt{2}}{2}=1\pm\sqrt{2}$

Answer:

$5, 1+\sqrt{2}, 1-\sqrt{2}$