Question

How do you factor completely \[8{{c}^{2}}-26c+15\]?

Answer 1

Hint: Consider the given equation as a quadratic equation in variable ‘c’. Now, use the middle term split method to factorize \[8{{c}^{2}}-26c+15\]. Split -26c into two terms in such a way that their sum is -26c and the product is \[120{{c}^{2}}\]. For this process, find the prime factors of 120 and combine them in such a way so that we can get our conditions satisfied. Finally, take common terms together and write \[8{{c}^{2}}-26c+15\] as a product of two terms.

Complete step by step answer:
Here, we have been asked to factorize \[8{{c}^{2}}-26c+15\]. Here, we can consider this equation as a quadratic equation in ‘c’.
Now, let us use the middle term split method for the factorization. It says that we have to split the middle terms which is -26c into two terms such that their sum is -26c and the product is equal to the product of constant term (15) and \[8{{c}^{2}}\], i.e., \[120{{c}^{2}}\]. To do this, first we need to find all the prime factors of 120. So, let us find.
We know that 120 can be written as: - \[120=2\times 2\times 2\times 3\times 5\] as the product of its primes. Now, we have to group these factors in such a way that our conditions of the middle term split method are satisfied. So, we have,
(i) \[\left( -6c \right)+\left( -20c \right)=-26c\]
(ii) \[\left( -6c \right)\times \left( -20c \right)=120{{c}^{2}}\]
Hence, both the conditions of the middle term split method are satisfied. So, the quadratic polynomial can be written as: -
\[\begin{align}
  & \Rightarrow 8{{c}^{2}}-26c+15=8{{c}^{2}}-6c-20c+15 \\
 & \Rightarrow 8{{c}^{2}}-26c+15=2c\left( 4c-3 \right)-5\left( 4c-3 \right) \\
\end{align}\]
Taking \[\left( 4c-3 \right)\] common in the R.H.S. we get,
\[\Rightarrow 8{{c}^{2}}-26c+15=\left( 4c-3 \right)\left( 2c-5 \right)\]
Hence, \[\left( 4c-3 \right)\left( 2c-5 \right)\] is the factored form of the given quadratic polynomial.

Note:
One may note that we can use another method for the factorization. The Discriminant method can also be applied to solve the question. What we will do is we will find the solution of the quadratic equation using discriminant method. The value of ‘c’ obtained will be written as c = m and c = n. Finally, we will consider the product \[\left( c-m \right)\left( c-n \right)\] to get the factored form.