Question

If one factor of \[{a^3} + {b^3}\] is \[\left( {a + b} \right)\] , then the other factor is
\[\left( 1 \right)\] \[{a^3} + {b^3} + ab\]
\[\left( 2 \right)\] \[a - b + ab\]
\[\left( 3 \right)\] \[{a^2} + {b^2} - ab\]
\[\left( 4 \right)\] \[{a^2} + {b^2} + ab\]

Answer 1

Hint: We have to find the other factor of the given expression \[{a^3} + {b^3}\] when one factor is given as \[\left( {a + b} \right)\]. We solve this question using the concept of expansion of the polynomial equations. We should have the knowledge of the formulas for the cube of the sum of two numbers , also the formula for the square of the sum of two numbers. First we will use the formula of a cube of sum of two numbers and then we will simplify the expression such that we get the formula as in terms of the given expression. And then using the factor given we will evaluate the other factor of the given expression.

Complete step-by-step solution:
Given :
One factor of \[{a^3} + {b^3}\] is \[\left( {a + b} \right)\]
As we know that the formula of cube of sum of two numbers is given as :
\[{(a + b)^3} = {a^3} + {b^3} + 3{a^2}b + 3a{b^2}\]
Now we can simplify the expression as according to the given expression as :
\[{a^3} + {b^3} = {(a + b)^3} - (3{a^2}b + 3a{b^2})\]
Now taking \[3ab\] common , we get the expression as :
\[{a^3} + {b^3} = {(a + b)^3} - 3ab(a + b)\]
Taking \[\left( {a + b} \right)\] common , we get the expression as :
\[{a^3} + {b^3} = [{(a + b)^2} - 3ab](a + b)\]
We also know that the formula for square of sum of two numbers is given as :
\[{(a + b)^2} = {a^2} + {b^2} + 2ab\]
Using the formula , we get the expression as :
\[{a^3} + {b^3} = [{a^2} + {b^2} + 2ab - 3ab](a + b)\]
On further simplifying , we get
\[{a^3} + {b^3} = [{a^2} + {b^2} - ab](a + b)\]
As , we know that one of the factor of \[{a^3} + {b^3}\] is \[\left( {a + b} \right)\] , then the other factor of the expression is \[[{a^2} + {b^2} - ab]\].
Thus , the correct option is \[\left( 3 \right)\].

Note: As we know, a given expression can be split into its factors by using the various formulas of polynomial expansion. Similarly if the given expression would have been \[{a^3} - {b^3}\] then we would have used the formula of the cube of the difference of the two numbers.