Question

# 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 isA. ${a^3} + {b^3} + {c^3}$B. ${a^2} + {b^2} - ab$C. ${a^2} + {b^2} + {c^2}$ D. ${a^2} + {b^2} + ab$

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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 ${a^2} + {b^2} + ab{\text{ is a factor of }}{a^4} + {b^4} + {a^2}{b^2}$

Now another factor is determined by following method:

${a^2} + {b^2} + ab\mathop{\left){\vphantom{1\begin{gathered} {a^4} + {b^4} + {a^2}{b^2} - {a^2}{\text{ - }}{a^2}{b^2}{\text{ - }}{a^3}b \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ {b^4}{\text{ }} - {a^3}b {b^4}{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2} \_\_\_\_\_\_\_\_\_\_\_\_\_\_ - {a^3}b{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2} - {a^3}b{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2} \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ }}0 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \end{gathered} }}\right. \!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered} {a^4} + {b^4} + {a^2}{b^2} - {a^2}{\text{ - }}{a^2}{b^2}{\text{ - }}{a^3}b \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ {b^4}{\text{ }} - {a^3}b {b^4}{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2} \_\_\_\_\_\_\_\_\_\_\_\_\_\_ - {a^3}b{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2} - {a^3}b{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2} \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ }}0 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \end{gathered} }}} \limits^{\displaystyle \,\,\, {{a^2} + {b^2} - ab}}$

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

Hence, the correct option is (b)

Note: - In these types of questions we simply divide the given factor with the given polynomial.