Question

x3_x2_(3_√3)x+√3 following polynomial s has (X+1) as a factor​

Answer 1

didn't understood the question clearly but answering the similar question

If (x+1) is a factor of given polynomial say p(X) , then at x= -1, p(x)  will become zero, otherwise it is not a factor of given polynomial.

 

Solution:

 

(i) x³+x²+x+1

 

Let p(x)= x³+x²+x+1

 

 

The zero of x+1 is -1.

 

On putting x= -1

p(−1)=(−1)³+(−1)²+(−1)+1

=−1+1−1+1=0

Hence, 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.

 

On putting x= -1

p(−1)=(−1)⁴+(−1)³+(−1)²+(−1)+1

 

=1−1+1−1+1=1≠0

 

Hence, 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.

 

On putting x= -1

p(−1)=(−1)⁴+3(−1)³+3(−1)²+(−1)+1

 

=1−3+3−1+1

=1≠0

 

Hence,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.

 

On Putting x= -1

p(−1)=(−1)³–(−1)²–(2+√2)(−1)+√2

Hence, by factor theorem, x+1 is not a factor of x³–x²–(2+√2)x+√2.