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Use the Remainder Theorem to determine which of the following is a factor of p(x) = x 3 - 2x 2 - 5x + 6.
a) x + 1
b) x - 2
c) x - 3
d) x + 3

Respuesta :

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.
kudzordzifrancis
ANSWER

Using the remainder theorem,
[tex]p( 3) = 0[/tex]
therefore
[tex]x - 3[/tex]

is a factor of the polynomial.

EXPLANATION

The given polynomial is
[tex]p(x) = {x}^{3} - 2 {x}^{2} - 5x + 6[/tex]

According to the remainder theorem if
[tex]p(a) = 0[/tex]
then,

[tex]x - a[/tex]
is a factor of the polynomial.

For the first option,

[tex]p( - 1) = {( - 1)}^{3} - 2 {( - 1)}^{2} - 5( - 1) + 6[/tex]

[tex]p( - 1) = - 1 - 2 + 5 + 6[/tex]

[tex]p( - 1) = 8[/tex]

Since
[tex]p( - 1) \ne0[/tex]

[tex]x + 1[/tex]
is not a factor of the polynomial.

For the second option,

[tex]p( 2) = {( 2)}^{3} - 2 {( 2)}^{2} - 5( 2) + 6[/tex]

[tex]p( 2) = 8 - 8 - 10+ 6[/tex]

[tex]p( 2) = - 4[/tex]

Since
[tex]p( 2) \ne0[/tex]
[tex]x - 2[/tex]
is also not a factor.

For the third option,



[tex]p( 3) = {( 3)}^{3} - 2 {( 3)}^{2} - 5( 3) + 6[/tex]

[tex]p( 3) = 27 - 18 - 15 + 6[/tex]

[tex]p( 3) = - 6 + 6[/tex]

[tex]p( 3) = 0[/tex]

Therefore
[tex]x - 3[/tex]
is a factor of the given polynomial.



For the last option,

[tex]p( - 3) = {( - 3)}^{3} - 2 {( - 3)}^{2} - 5( - 3) + 6[/tex]

[tex]p( - 3) = - 27- 18 + 15+ 6[/tex]

[tex]p( - 3) = - 24[/tex]
This is again not a factor.


The correct answer is C.

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