Question

determine whether( X + 2) is a factor of 2 X ^ 4 + X ^ 3 + 4 x ^ 2 - X - 7.
please answer it with every steps!!

Answer 1

According to The Remainder Theorem  

(

x

−

a

)

is a factor of a polynomial  

P

(

x

)

if and only if  

P

(

a

)

=

0

.

So to check if  

(

x

−

1

)

is a factor of  

P

(

x

)

=

4

x

4

−

2

x

3

+

3

x

2

−

2

x

+

1

you have to check if  

P

(

1

)

=

0

P

(

1

)

=

4

⋅

1

4

−

2

⋅

1

3

+

3

⋅

1

2

−

2

⋅

1

+

1

=

4

−

2

+

3

−

2

+

1

P

(

1

)

=

4

P

(

1

)

is not zero, so  

(

x

−

1

)

is not the factor of  

P

(

x

)

In fact  

P

(

1

)

=

4

means that the remainder when  

4

x

4

−

2

x

3

+

3

x

2

−

2

x

+

1

is divided by  

x

−

1

is  

4

Answer 2

Hello ☺️☺️

_______________________

Given that ;

G(x) = x + 2

⇒ x = -2

P(x) = 2x⁴ + x³ + 4x² - x - 7

⇒ 2 (-2)⁴ + (-2)³ + 4(-2)² - (-2) - 7

⇒ 2 × 16 - 8 + 16 + 2 - 7

⇒ 32 + 2 + 16 - 8 - 7

⇒ 50 - 15

⇒ 35

Hence,

( x + 2 ) is not the factor of 2x⁴ + x³ + 4x² - x - 7.

_______________________

Thanks !

@AarohiG ❤️