Question

Determine which of the following polynomials has (x + 1) a factor :
(1) r + x + x + 1
(ii) x + x + x² + x + 1
(iii) x4 + 3x3 +3x2+ x + 1 (iv) x3-x2 (2+ 2)x+ V2​

Answer 1

Answer:

Apply remainder theorem

x+1=0

x=−1

Put the value of x=−1 in all equations.

(i) x3 +x2+x+1=(−1)3 +(−1)2 +(−1)+1=−1+1−1+1=0

Then x+1 is the factor of equation

(ii) x 4 +x3+x2 +x+1=(−1)4 +(−1)3 +(−1)2+(−1)+1=1−1+1−1+1=1

This is not zero.Then x+1 is not the factor of equation

(iii) x4 +3x3+3x2 +x+1=(−1)4+3(−1)3 +3(−1)2+(−1)+1=1

This is not zero.Then x+1 is not the factor of equation

(iv)x3 −x 2 −(2+√2 )x+√2=(−1)3−(−1)2 −(2+√2 )(−1)+√2

=−1−1+2−√2+√2 =0

This is zero. Then x+1 is the factor of equation.

Hope this is helpful for you.