Question

select all zeros of the polynomial function.
$f(x) = x^4 - 6x^3 + 7x^2 + 6x - 8$

Explanation:

Step1: Test x=1 via Remainder Theorem

$f(1)=1^4 - 6(1)^3 +7(1)^2 +6(1)-8=1-6+7+6-8=0$

Step2: Factor out $(x-1)$ via division

Use polynomial long division or synthetic division:
$\frac{x^4 -6x^3 +7x^2 +6x-8}{x-1}=x^3-5x^2+2x+8$

Step3: Test x=4 on cubic polynomial

$f(4)=4^3 -5(4)^2 +2(4)+8=64-80+8+8=0$

Step4: Factor out $(x-4)$ from cubic

$\frac{x^3-5x^2+2x+8}{x-4}=x^2-x-2$

Step5: Factor quadratic polynomial

$x^2-x-2=(x-2)(x+1)$

Step6: Identify all zeros

Set each factor equal to 0: $x-1=0$, $x-4=0$, $x-2=0$, $x+1=0$

Answer:

-1, 1, 2, 4