Question

Determine which of the following polynomials has (x + 1) a factor :
(0) x3+ x2 + x + 1
(ii) x4+ x3 + x2 + x + 1​

Answer 1

Step-by-step explanation:

= x³+x²+x+1

=> x+1=0

x=-1

(-1)³+(-1)²+(-1)+1

-1 +1 -1 +1

2-2 = 0

so that x³+x²+x+1 is a factor of x+1

Answer 2

The zero of x + 1 is -1.

(i) Let p (x) = x3 + x2 + x + 1

∴ p (-1) = (-1)3 + (-1)2 + (-1) + 1 .

= -1 + 1 – 1 + 1

⇒ p (- 1) = 0

So, (x+ 1) is a factor of x3 + x2 + x + 1.

(ii) Let p (x) = x4 + x3 + x2 + x + 1

∴ P(-1) = (-1)4 + (-1)3 + (-1)2 + (-1)+1

= 1 – 1 + 1 – 1 + 1

⇒ P (-1) ≠ 1

So, (x + 1) is not a factor of x4 + x3 + x2 + x+ 1.