 # Find the complex zeros of the following polynomial function. Write f in factored form.

2021-10-18

Find the complex zeros of the following polynomial function. Write f in factored form.

$$f(x)=x^3-8$$

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Given $$f(x)=x^3-8$$

Set $$x^3-8$$ equal to $$0$$

Solve for $$x$$.

Add $$8$$ to both sides of the equation.

$$x^3=8$$

Subtract $$8$$ from both sides of the equation.

$$x^3−8=0$$

Factor the left side of the equation

$$(x−2)(x^2+2x+4)=0$$

If any individual factor on the left side of the equation is equal to $$0$$, the entire expression will be equal to $$0$$.

$$x−2=0$$

$$x^2+2x+4=0$$

Set $$x−2$$ equal to $$0$$ and solve for $$x$$.

$$x=2$$

Set $$x^2+2x+4$$ equal to $$0$$ and solve for $$x$$.

$$x=−1+i \sqrt3,−1−i \sqrt3$$

The final solution is all the values that make $$(x−2)(x^2+2x+4)=0$$ true

Answer: $$x=2,−1+i \sqrt3,−1−i \sqrt3$$