Question

H.C.F of $x^3-1$ and $x^4+x^2+1$ will be

• A. $x-1$
• B. $x^2+x+1$
• C. $x^2-x+1$
• D. $x^2-x-1$

Answer 1

Answer: (B)

$x^2+x+1$

H.C.F of $x^3-1\&x^4+x^2+1$

H.C.F is defined as the highest common factor of two/more numbers.

Here we need to factorize both the polynomial and find the common factor among them.

$\Rightarrow x^3-1=x^3-{\left(1\right)}^3=(x-1)(x^2+x+1)$ ( using $a^3-b^3=(a-b)(a^2+ab+b^2)$ )

$\Rightarrow x^4+x^2+1=x^4+2x^2+1-x^2={(x^2+1)}^2-{\left(x\right)}^2=(x^2+1-x)(x^2+x+1)$ ( using $a^2-b^2=(a+b)(a-b)$ )

$\therefore$ The common factor among the polynomial is $(x^2+x+1).$

Hence, the answer is $(x^2+x+1).$