Question

How do you factor the trinomial $3{{x}^{2}}+11x+6$?

Answer 1

Hint: In this problem we need to calculate the factors of the given equation. We can observe that the given equation is quadratic equation or trinomial equation which is in the form of $a{{x}^{2}}+bx+c$. So, we will compare the given equation with $a{{x}^{2}}+bx+c$ and write the values of $a$, $b$, $c$. Now we will calculate the value of $ac$ and write the factors of the calculated value. From the factors of $ac$, we will select any two factors such that ${{x}_{1}}+{{x}_{2}}=b$. By using these values, we will split the middle term in the given equation and take appropriate terms as common to get the required solution.

Complete step by step solution:
Given equation, $3{{x}^{2}}+11x+6$.
Comparing the above equation with $a{{x}^{2}}+bx+c$, then we will get
$a=3$, $b=11$, $c=6$.
Now the value of $ac$ will be $ac=3\times 6=18$ and the factors of $18$ are $1$, $2$, $3$, $6$, $9$, $18$. From these factors, we can write that
$\begin{align}
  & 2\times 9=18 \\
 & 2+9=11 \\
\end{align}$
So, splitting the middle term $11x$ as $2x+9x$, then the given equation is modified as
$\Rightarrow 3{{x}^{2}}+11x+6=3{{x}^{2}}+2x+9x+6$
Taking $x$ as common from the terms $3{{x}^{2}}$, $2x$ and $3$ from $9x$, $6$, then we will get
$\Rightarrow 3{{x}^{2}}+11x+6=x\left( 3x+2 \right)+3\left( 3x+2 \right)$
Now taking $3x+2$ as common from the above equation, then we will have
$\Rightarrow 3{{x}^{2}}+11x+6=\left( 3x+2 \right)\left( x+3 \right)$
Hence the factors of the given equation $3{{x}^{2}}+11x+6$ are $3x+2$, $x+3$. The graph of the given polynomial will be

Note: The above followed method is also used to calculate the roots of the quadratic equation and it is called as factorization method. After calculating the factors of the given equation, we will equate them individually to zero and simplify them to get the roots of the given quadratic equation.