Question

select all zeros of the polynomial function. \\( f(x) = x^4 + 5x^3 - 7x^2 - 29x + 30 \\)

Explanation:

Step1: Test x=-5 via Remainder Theorem

$f(-5)=(-5)^4 + 5(-5)^3 -7(-5)^2 -29(-5)+30=625-625-175+145+30=0$

Step2: Factor out (x+5) via division

Use polynomial division: $\frac{x^4+5x^3-7x^2-29x+30}{x+5}=x^3-7x+6$

Step3: Test x=1 on cubic polynomial

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

Step4: Factor out (x-1) from cubic

Polynomial division: $\frac{x^3-7x+6}{x-1}=x^2+x-6$

Step5: Factor quadratic polynomial

$x^2+x-6=(x+3)(x-2)$

Step6: Find all zeros

Set each factor to 0: $x+5=0, x-1=0, x+3=0, x-2=0$

Answer:

-5, 1, -3, 2