Question

Select all of the roots of x3 + 2x2 – 16x– 32.

–1

–2

–4

1

4

5

Answer 1

-2, -4, and 4 are the correct answers on edge

Explanation:

Answer 2

-4, -2, 4

Explanation:

You want the roots of the polynomial x³ +2x² -16x -32.

Factors

The polynomial can be factored by grouping:

x³ +2x² -16x -32

= (x³ +2x²) -(16x +32) . . . . . . group pairs of terms

= x²(x +2) -16(x +2) . . . . . . . . factor each group

= (x² -16)(x +2) . . . . . . . . . . . . factor out the common factor

Now, the difference of squares can be factored:

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

Roots

The zero product rule tells us the roots are the values of x that make the factors zero:

x -4 = 0 ⇒ x = 4

x +4 = 0 ⇒ x = -4

x +2 = 0 ⇒ x = -2

The roots of the polynomial are {-4, -2, 4}.

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Additional comment

The factorization of the difference of squares is a special form:

a² -b² = (a +b)(a -b)