Question

select all of the roots of \\(x^3 + 2x^2 - 16x - 32\\).
-8
-2
-4
2
4
8

Explanation:

🆕 New Concept Discovered: Finding Roots of Polynomials by Grouping
Breaking a four-term polynomial into two pairs to find common factors.

Step 1: Group the terms

To find the roots of the polynomial equation \( x^3 + 2x^2 - 16x - 32 = 0 \), we can group the terms into two pairs:

$$ (x^3 + 2x^2) - (16x + 32) = 0 $$

Step 2: Factor out the greatest common factor (GCF) from each group

From the first group \( (x^3 + 2x^2) \), we factor out \( x^2 \):

$$ x^2(x + 2) $$

From the second group \( -(16x + 32) \), we factor out \( -16 \):

$$ -16(x + 2) $$

Now, combine them:

$$ x^2(x + 2) - 16(x + 2) = 0 $$

Step 3: Factor out the common binomial

Both terms share a common binomial factor of \( (x + 2) \). Factoring it out gives:

$$ (x^2 - 16)(x + 2) = 0 $$

Step 4: Factor the difference of squares

The term \( x^2 - 16 \) is a difference of squares, which factors into \( (x - 4)(x + 4) \):

$$ (x - 4)(x + 4)(x + 2) = 0 $$

Step 5: Solve for the roots

Set each factor equal to zero to find the values of \( x \):

$$ x - 4 = 0 \implies x = 4 $$

$$ x + 4 = 0 \implies x = -4 $$

$$ x + 2 = 0 \implies x = -2 $$

The roots of the polynomial are \( -4 \), \( -2 \), and \( 4 \).

Answer:

The correct choices to select are:

• -4
• -2
• 4