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 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.

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:

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

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.