Question

(x+2) is a factor of p(x)=x³+x²+x+2,Examine the validity of the given statement.

Answer 1

Given, p(x) = x³ + x² + x + 2.
For (x + 2) to be a factor of p(x), the condition: p(–2) = 0; should be satisfied.

p(–2) = (–2)³ + (–2)² + (–2) + 2

= (–8) + (4) + (–2) + 2

= – 8 + 4 – 2 + 2

= –4 ≠ 0

Thus, (x + 2) is not a factor of the given polynomial.

therefore, the given statement is invalid.

Answer 2

Hi ,

It is given that ,

p( x ) = x³ + x² + x + 2 ,

Let g( x ) = x + 2 ,

The zero of g( x ) is -2 .

Then p( -2 ) = ( -2 )³ + ( -2 )² + ( -2 ) + 2

= -8 + 4 - 2 + 2

= - 10 + 6

= - 4

≠ 0

So , by the Factor Theorem ,

x + 2 is not a factor of p( x ).

I hope this helps you.

: )