Question

Solve ${m^5} + {m^4} + {m^3} + {m^2} + m + 1 = ({m^3} + 1) \times \_\_\_\_\_\_\_\_\_\_\_\_\_$
A. ${m^5} + {m^4} + {m^2} + m$
B. ${m^2} + {m^3}$
C. ${m^3} + {m^3} + m + 1$
D. ${m^2} + m + 1$

Answer 1

Hint: Find the common factor in the original term. Decide the common factor based on terms in the first bracket on the right hand side.

Complete step-by-step answer:
There are six terms in the left hand side of the equation and the first term in the bracket on the right hand side is ${m^3}$. So first we will gather the first three terms of left hand side since the third term is ${m^3}$. We get the following expression:
$ \Rightarrow {m^3}({m^2} + m + 1) + {m^2} + m + 1$
Here, we have used the following relation to find the common factor.
$ \to {x^a} \times {x^b} = {x^{a + b}}$
Now, we see that three terms in the bracket are equal to the last three terms of the last expression. Taking the three terms as common, we get,
$ \Rightarrow ({m^2} + m + 1)({m^3} + 1)$
Comparing above expression with the expression in question, we conclude that $({m^2} + m + 1)$ should fill the blank. Thus, option D is the correct option.

Note: The other method of obtaining the solution is by considering the answer as y and then dividing both sides by $({m^3} + 1)$. Then the quotient of the division is equal to y.