Question

Determine which of the following polynomials has (x+1) as a factor.
(ii) x4 - x3 + x2 - x +1
(iii) x4 + 2x3 + 2x2 + x + 1
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Answer 1

Answer:

x+1 is factor then x = -1 should be zero of polynomial. now putting x= -1 , we get

$(ii) \: \: ( - 1) {}^{4} - ( - 1) {}^{3} + ( - 1) {}^{2} - ( - 1) + 1 = 1 + 1 + 1 + 1 + 1 = 5 \: not \: equal \: to \: zero \: so \: (x + 1) \: is \: not \: factor.$

now,

same process in (iii) ..

Answer 2

The zero of x + 1 is -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.

(iii) Let p (x) = x4 + 3x3 + 3x2 + x + 1 .

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

= 1 – 3 + 3 – 1 + 1 = 1

⇒ p (-1) ≠ 0

So, (x + 1) is not a factor of x4 + 3x3 + 3x2 + x+ 1.