Question

Divide \[{x^4} - {y^4}\] by \[x + y\].

Answer 1

Hint:
Here we will use the basic concept of the division. First, we will use the algebraic identity to simplify the equation \[{x^4} - {y^4}\]. Then we will apply the division operation and divide the simplified equation by \[x + y\] to get the required value.

Complete step by step solution:
We have to divide \[{x^4} - {y^4}\] by \[x + y\].
First, we will break the exponent of \[{x^4} - {y^4}\]. Therefore, we get
\[{x^4} - {y^4} = {\left( {{x^2}} \right)^2} - {\left( {{y^2}} \right)^2}\]
Now applying the algebraic identity i.e. \[{a^2} - {b^2} = (a - b)(a + b)\], we get
\[ \Rightarrow {x^4} - {y^4} = \left( {{x^2} - {y^2}} \right)\left( {{x^2} + {y^2}} \right)\]
Now we will again apply the same algebraic identity on \[\left( {{x^2} - {y^2}} \right)\]. Therefore , we get
\[ \Rightarrow {x^4} - {y^4} = \left( {x + y} \right)\left( {x - y} \right)\left( {{x^2} + {y^2}} \right)\]
Now we will divide this simplified equation by \[x + y\]. Therefore, we get
\[\begin{array}{l} \Rightarrow \dfrac{{{x^4} - {y^4}}}{{x + y}} = \dfrac{{\left( {x + y} \right)\left( {x - y} \right)\left( {{x^2} + {y^2}} \right)}}{{x + y}}\\ \Rightarrow \dfrac{{{x^4} - {y^4}}}{{x + y}} = \left( {{x^2} + {y^2}} \right)\left( {x - y} \right)\end{array}\]

Hence, \[{x^4} - {y^4}\] divided by \[x + y\] is equals to \[\left( {{x^2} + {y^2}} \right)\left( {x - y} \right)\].

Additional Information:
The division is the operation in which the dividend is divided by the divisor to get the quotient along with some remainder. So, the general formula of the division operation is
\[{\rm{Dividend}} = \left( {{\rm{Divisor}} \times {\rm{Quotient}}} \right) + {\rm{Remainder}}\]
Dividend is the term or number which is to be divided. Divisor is the term or number which we divide by. Quotient is the term or number which is the answer of this division operation and remainder is the term which is left when a division operation is performed.

Note:
We know that addition is the operation in which two numbers are combined to get the result. Subtraction is the operation which gives us the difference between the two numbers. Multiplication is the operation in which the one number is added to itself for some particular number of times.
Algebraic identities are equations where the value of the left-hand side of the equation is identically equal to the value of the right-hand side of the equation.
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
\[{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\]