Question

determine the number of zeros of the polynomial function.
$f(x) = x^2 + 3x - 4$
_______ zero(s)
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1. - / 1 points

determine the number of zeros of the polynomial function.
$f(x) = x^5 - x^6$
_______ zero(s)
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find all real solutions of the polynomial equation. (enter your answers as a comma-separated list.)
$x^4 - 7x^2 - 6x = 0$
$x =$ _______
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Explanation:

Step1: Identify degree of polynomial 1

The polynomial $f(x)=x^2+3x-4$ is degree 2.

Step2: Apply Fundamental Theorem of Algebra

A degree $n$ polynomial has $n$ complex zeros.

Step3: Identify degree of polynomial 2

Rewrite $f(x)=x^5-x^6$ as $f(x)=-x^6+x^5$, degree 6.

Step4: Apply Fundamental Theorem of Algebra

Degree 6 polynomial has 6 complex zeros.

Step5: Factor polynomial equation 3

Factor out $x$: $x(x^3-7x-6)=0$.

Step6: Factor cubic polynomial

Factor $x^3-7x-6$: $(x+1)(x+2)(x-3)=0$.

Step7: Find all real solutions

Set each factor equal to 0.

Answer:

1. 2 zero(s)
2. 6 zero(s)
3. $x = -2, -1, 0, 3$