Question

The polynomial x^2 +3x+1 is a factor of x^3+2x^2-5x-6 Please select the best answer from the choices provided

True
False

Answer 1

Answer: False

x^3+2x^2-5x-6 factors to (x-2)(x+1)(x+3)

None of the pair of factors multiply to x^2+3x+1

(x-2)(x+1) = x^2-x-2
(x-2)(x+3) = x^2+x-6
(x+1)(x+3) = x^2+4x+3

Answer 2

A factor of a polynomial evenly divides the polynomial, without leaving a

remainder.

• The statement, the polynomial x² + 3·x + 1 is a factor of x³ + 2·x² - 5·x - 6 is; False

Reasons:

By long division we have;

$\displaystyle {}\hspace{2.1cm} x - 1\\x^2 + 3 \cdot x + 1|\overline{x^3 + 2 \cdot x^2 - 5 \cdot x - 6}\\{}\hspace{2.1cm} x^3 + 3\cdot x^2 + x\\{}\hspace{2.5cm} -x^2 - 6 \cdot x\\{}\hspace{2.5cm} - x^2-3 \cdot x- 1\\{}\hspace{3.3cm} -3 \cdot x - 5 (remainder)$

Given that (x³ + 2·x² - 5·x - 6) ÷ (x² + 3·x + 1) = (x - 1) remainder (-3·x - 5), we have;

(x² + 3·x + 1) is not a factor of (x³ + 2·x² - 5·x - 6) because it does not evenly divides (x³ + 2·x² - 5·x - 6)

Therefore, the polynomial, x² + 3·x + 1 is not a factor of x³ + 2·x² - 5·x - 6, the

statement is false

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