Question

How do you factor completely $10{{x}^{2}}-15xy$?

Answer 1

Hint: The given polynomial $10{{x}^{2}}-15xy$ consists of only two terms. So in order to factor it, we can simply take the highest factor common to both the terms outside it. We can clearly observe that the factor of $x$ is common to both the terms. So $x$ can be taken outside of the polynomial to factorize the polynomial. Then the factor of \[5\] must be taken outside so as to completely factorize the given polynomial.

Complete step by step solution:
Let us call the polynomial given in the above question as $p\left( x,y \right)$ so that we can write the below equation
$\Rightarrow p\left( x,y \right)=10{{x}^{2}}-15xy$
The factor of $x$ is common to both the terms in the above polynomial. So we can take it outside to get
$\Rightarrow p\left( x,y \right)=x\left( 10x-15y \right)$
Now, we can see that in the second factor $\left( 10x-15y \right)$ of the above polynomial, the factor of $5$ is common to both of its terms. Therefore, we can take it outside so that we can rewrite the above polynomial as
$\Rightarrow p\left( x,y \right)=5x\left( 2x-3y \right)$
Now, no factor is common in the above polynomial. So we can say that we have factorized the given polynomial completely.
Hence, the given polynomial is factored completely as $5x\left( 2x-3y \right)$.

Note:
The polynomial which is given to us in the above question is a polynomial in two variables. Do not consider it to be a quadratic polynomial by seeing the second degree of the variable $x$. In such two degree polynomials, the method of taking the common factors outside is the only possible method for the factorization.