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question 1
select all x-coordinates of x-intercepts of the graph of the function ( f(x) = x^3 + 2x^2 + 8x + 16 ).
-2
nothing in this list is correct.
± 8
2
( pm 2sqrt{2} )

Explanation:

Step1: Factor the function

To find the x - intercepts, we set \( f(x)=0 \), so we have the equation \( x^{3}+2x^{2}+8x + 16=0 \). We can factor this polynomial by grouping.
Group the first two terms and the last two terms:
\( (x^{3}+2x^{2})+(8x + 16)=0 \)
Factor out the greatest common factor from each group:
\( x^{2}(x + 2)+8(x + 2)=0 \)
Now, factor out the common binomial factor \( (x + 2) \):
\( (x^{2}+8)(x + 2)=0 \)

Step2: Solve for x

We set each factor equal to zero and solve for x.

• For the factor \( x + 2=0 \), we get \( x=-2 \).
• For the factor \( x^{2}+8 = 0 \), we have \( x^{2}=-8 \), and the solutions for x are \( x=\pm\sqrt{-8}=\pm2i\sqrt{2} \), which are complex numbers. Since we are looking for x - intercepts (real - valued x where the graph crosses the x - axis), we only consider the real solution from the factoring.

Answer:

-2