Question

How do you factor $5{{a}^{2}}-25a-a+5$ ?

Answer 1

Hint: Now in the given expression we will first take 5a common from the first two terms. Further we will take – 1 common from the last two terms. Hence we will further simplify the expression by using distributive property. Hence we will factor the given expression.

Complete step-by-step answer:
Now consider the given expression $5{{a}^{2}}-25a-a+5$
We know that the given expression is a quadratic expression in a.
Now we can see that the middle term in the expression is split such that the product of the first term and the last term is equal to the product of first term and the last term.
Hence we can factor the expression given easily by taking the terms common.
Now let us take 5a common from the first two terms in the expression. Hence we get the given expression as, $5a\left( a-5 \right)-a+5$
Now let us take – 1 common from the last two terms in the expression. Then we get,
$\Rightarrow 5a\left( a-5 \right)-1\left( a-5 \right)$
Now we know that the distributive property states that $ac-bc=\left( a-b \right)c$ . Hence using this property we get the given expression as,
$\Rightarrow \left( 5a-1 \right)\left( a-5 \right)$
Hence the given expression is factorized and $\left( 5a-1 \right)$ and $\left( a-5 \right)$ are the factors of the given expression.

Note: Now note that for any quadratic expression in general form we can split the middle term such that the product of the terms is the product of the first term and last term. Hence we can further easily solve the expression for factors. Now if such splitting is not possible we can find the roots of the expression and find corresponding factors.