Question

Factorize : $a^6-b^6$ .

• A. $(b-a)(a+b)(a^3+ab+b^3)(a^2-ab+b^2)$
• B. $(a-b)(a+b)(a^2+ab+b^2)(a^2-ab+b^2)$
• C. $(a+b)(a+b)(a^2+ab+b^2)(2a^2-ab+b^2)$
• D. $(a-b)(a+b)(a^2+ab+b^2)(a^3-ab+b^3)$

Answer 1

The correct option is B

$(a-b)(a+b)(a^2+ab+b^2)(a^2-ab+b^2)$

Factorize the given polynomial

$$\begin{gathered} a^6-b^6 \\ {\left(a^3\right)}^2-{\left(b^3\right)}^2 \end{gathered}$$

Use the identity ${\mathit{x}}^{\mathbf{2}}\mathbf{-}{\mathit{y}}^{\mathbf{2}}\mathbf{=}\mathbf{(}\mathit{x}\mathbf{-}\mathit{y}\mathbf{)}\mathbf{(}\mathit{x}\mathbf{+}\mathit{y}\mathbf{)}$

$a^6-b^6=(a^3-b^3)(a^3+b^3)$

Use the identity ${\mathit{x}}^{\mathbf{3}}\mathbf{-}{\mathit{y}}^{\mathbf{3}}\mathbf{=}\mathbf{(}\mathit{x}\mathbf{-}\mathit{y}\mathbf{)}\mathbf{(}{\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{+}\mathit{x}\mathit{y}\mathbf{)}$ and ${\mathit{x}}^{\mathbf{3}}\mathbf{+}{\mathit{y}}^{\mathbf{3}}\mathbf{=}\mathbf{(}\mathit{x}\mathbf{+}\mathit{y}\mathbf{)}\mathbf{(}{\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{-}\mathit{x}\mathit{y}\mathbf{)}$

$a^6-b^6=(a-b)(a^2+b^2+ab)(a+b)(a^2+b^2-ab)$

Hence the factors of the polynomial $a^6-b^6$are $(a-b),(a+b),(a^2+b^2-ab)$ and$(a^2+b^2+ab)$.

Therefore, option B is correct.