Question

How do you factor the trinomial $6{{x}^{2}}+37x+6$?

Answer 1

Hint: Factorization is the process of obtaining factors of a number that is the numbers that divide this number. Factorization writes a number as the product of smaller numbers. Factorization is the process of reducing the bracket of a reducing the bracket of a quadratic equation, instead of expanding the bracket and converting the equation to a product of factors which cannot be reduced further. There are many methods for the factorization process. For the given question we have to follow the new AC method which is explained below. For a polynomial of the form $a{{x}^{2}}+bx+c$, rewrite the middle term as a sum of two terms whose product is $a.c$

Complete step by step answer:
From the given question, given expression is, $6{{x}^{2}}+37x+6$
As we can clearly observe that the given trinomial is in the form of $a{{x}^{2}}+bx+c$
By comparing the coefficients of both the equations we get the values of the variables,
$\begin{align}
  & a=6 \\
 & b=37 \\
 & c=6 \\
\end{align}$
Now, as we have been already discussed above, rewrite the middle term as sum of two terms and their product is $a.c$
$a.c=6.6=36$
$b=37$
Now, we can rewrite the equation as, $6{{x}^{2}}+x+36x+6$
Now, by factoring out the greatest common factor from each group we get $x\left( 6x+1 \right)+6\left( 6x+1 \right)$
$\Rightarrow \left( 6x+1 \right)\left( x+6 \right)$

Now we can conclude that $6x+1$ and $x+6$ are the factors of $6{{x}^{2}}+37x+6$

Note: We should be well known about the process of the factorization. We should be careful while splitting the constant into the product of the two numbers. We should be careful that the two numbers sum must be equal to the coefficient of the variable of degree one. Here we can also obtain factors by using the zeroes of the quadratic expression $6{{x}^{2}}+37x+6$ using the formulae $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ we will have $\begin{align}