Question

Use the Remainder Theorem to determine which of the following is a factor of p(x) = x 3 - 2x 2 - 5x + 6.

Answer 1

If x + 1 is a factor then f(-1) will = zero.

f(-1) =  8  so its not answer a).

f(3)  = 0   so the answer is x - 3

Its c.

Answer 2

ANSWER

Using the remainder theorem,
$p( 3) = 0$
therefore
$x - 3$

is a factor of the polynomial.

EXPLANATION

The given polynomial is
$p(x) = {x}^{3} - 2 {x}^{2} - 5x + 6$

According to the remainder theorem if
$p(a) = 0$
then,

$x - a$
is a factor of the polynomial.

For the first option,

$p( - 1) = {( - 1)}^{3} - 2 {( - 1)}^{2} - 5( - 1) + 6$

$p( - 1) = - 1 - 2 + 5 + 6$

$p( - 1) = 8$

Since
$p( - 1) \ne0$

$x + 1$
is not a factor of the polynomial.

For the second option,

$p( 2) = {( 2)}^{3} - 2 {( 2)}^{2} - 5( 2) + 6$

$p( 2) = 8 - 8 - 10+ 6$

$p( 2) = - 4$

Since
$p( 2) \ne0$
$x - 2$
is also not a factor.

For the third option,



$p( 3) = {( 3)}^{3} - 2 {( 3)}^{2} - 5( 3) + 6$

$p( 3) = 27 - 18 - 15 + 6$

$p( 3) = - 6 + 6$

$p( 3) = 0$

Therefore
$x - 3$
is a factor of the given polynomial.



For the last option,

$p( - 3) = {( - 3)}^{3} - 2 {( - 3)}^{2} - 5( - 3) + 6$

$p( - 3) = - 27- 18 + 15+ 6$

$p( - 3) = - 24$
This is again not a factor.


The correct answer is C.