Determine which of the following polynomials has (x + 1) a factor:
(i) x3 + x2 + x + 1
(ii) x4 + x3 + x2 + x + 1
(iii) x4 + 3x3 + 3x2 + x + 1
(iv) x3 – x2– (2+√2)x +√2
(i) x3+x2+x+1
Let p(x) = x3+x2+x+1
The zero of x+1 is -1. [x+1 = 0 means x = -1]
p(−1) = (−1)3+(−1)2+(−1)+1
= −1+1−1+1
= 0
∴By factor theorem, x+1 is a factor of x3+x2+x+1
(ii) x4+x3+x2+x+1
Let p(x)= x4+x3+x2+x+1
The zero of x+1 is -1. . [x+1= 0 means x = -1]
p(−1) = (−1)4+(−1)3+(−1)2+(−1)+1
= 1−1+1−1+1
= 1 ≠ 0
∴By factor theorem, x+1 is not a factor of x4 + x3 + x2 + x + 1
(iii) x4+3x3+3x2+x+1
Let p(x)= x4+3x3+3x2+x+1
The zero of x+1 is -1.
p(−1)=(−1)4+3(−1)3+3(−1)2+(−1)+1
=1−3+3−1+1
=1 ≠ 0
∴By factor theorem, x+1 is not a factor of x4+3x3+3x2+x+1
(iv) x3 – x2– (2+√2)x +√2
Let p(x) = x3–x2–(2+√2)x +√2
The zero of x+1 is -1.
p(−1) = (-1)3–(-1)2–(2+√2)(-1) + √2 = −1−1+2+√2+√2
= 2√2 ≠ 0
∴By factor theorem, x+1 is not a factor of x3–x2–(2+√2)x +√2