Question

Factorize the following equation: $x{y^4} + {x^4}y$

Answer 1

Hint: We will take the common parts from both the terms. We will use the formula $\left( {{a^3} + {b^3}} \right) = \left( {a + b} \right)\left( {{a^2} + {b^2} - ab} \right)$ and expand our equation. We should be familiar with algebraic identities like \[\,{(a + b)^2} = {a^2} + 2ab + {b^2}\] , \[{a^2} - {b^2} = (a - b)(a + b)\] etc.

Complete step-by-step solution:
Factorization is the term used to describe the process of forming factors. Factorization is the process of writing a number in the form of its component factors, which when multiplied together yield the original number.
We have to factorize $x{y^4} + {x^4}y$
$ = x{y^4} + {x^4}y$
We will take x and y common from each term, we get
$ = xy\left( {{y^3} + {x^3}} \right)$
We know that $\left( {{a^3} + {b^3}} \right) = \left( {a + b} \right)\left( {{a^2} + {b^2} - ab} \right)$
So, we will expand the term $\left( {{y^3} + {x^3}} \right)$ using the above formula
$ = xy\left( {y + x} \right)\left( {{x^2} + {y^2} - xy} \right)$
Hence, the factorization of $x{y^4} + {x^4}y = xy\left( {y + x} \right)\left( {{x^2} + {y^2} - xy} \right)$ .

Note: It's important to remember that in these types of questions, basic algebraic identities can be applied to solve and identify the equation's components. Algebraic identities are equations in which the value of the left-hand side of the equation is the same as the value of the right-hand side. In most cases, algebraic identities are employed to answer these types of problems.