If one factor of ${a^4} + {\text{ }}{b^4} + {\text{ }}{a^2}{b^2}$ is ${a^2} + {\text{ }}{b^2} + {\text{ }}ab$, then the other factor is:

${\text{A}}{\text{. }}{a^3} + {b^3} + {c^3}$

$ {\text{B}}{\text{. }}{a^2} + {b^2} - ab $

$ {\text{C}}{\text{. }}{a^2} + {b^2} + {c^2} $

$ {\text{D}}{\text{. }}{{\text{a}}^2}{\text{ + }}{{\text{b}}^2}{\text{ + ab}} $

Hint: Use long division method to divide ${a^4} + {b^4} + {a^2}{b^2}$ by the given factor ${a^2} + {b^2} + ab$ to find other factor.

Complete step by step answer:

We know that, if we are provided with a factor of a number and we have to find another factor we simply divide the number by the given factor.

For example we know that 3 is one of a factor of 12 and we have to find another factor we simply divide 12 by 3 to know another factor i.e.

$ \dfrac{{12}}{3} = 4 $

Thus, 4 is another factor of 12.

In the similar way we will find the factor of given question

Given that ${a^2} + {b^2} + ab$ is a factor of ${a^4} + {b^4} + {a^2}{b^2}$

Now another factor is determined by long division method:

$ {a^2} + {b^2} + ab\mathop{\left){\vphantom{1\begin{gathered}

{a^4} + {b^4} + {a^2}{b^2}

- {a^2}{\text{ - }}{a^2}{b^2}{\text{ - }}{a^3}b

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

{b^4}{\text{ }} - {a^3}b

{b^4}{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2}

\_\_\_\_\_\_\_\_\_\_\_\_\_\_

- {a^3}b{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2}

- {a^3}b{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2}

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

{\text{ }}0

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

\end{gathered} }}\right.

\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}

{a^4} + {b^4} + {a^2}{b^2}

- {a^2}{\text{ - }}{a^2}{b^2}{\text{ - }}{a^3}b

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

{b^4}{\text{ }} - {a^3}b

{b^4}{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2}

\_\_\_\_\_\_\_\_\_\_\_\_\_\_

- {a^3}b{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2}

- {a^3}b{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2}

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

{\text{ }}0

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

\end{gathered} }}}

\limits^{\displaystyle \,\,\, {{a^2} + {b^2} - ab}} $

Thus the another factor is ${a^2} + {b^2} - ab$

$ \therefore$ The correct option is (b)

Note: - When one factor of the polynomial is given and asked us to find another polynomial we will factorise the given polynomial or use a long division method to find the solution.