Question

How do you factor $3{{x}^{2}}+28x+32$?

Answer 1

Hint: First multiply the coefficient of \[{{x}^{2}}\] with the constant term. Then find two numbers whose sum or difference is the coefficient of ‘x’ and multiplication is the previous result. Split the given equation in terms of multiplication of two factors. Do the necessary simplification to obtain the required result.

Complete step by step solution:
Factorization: It is simply the resolution of a polynomial into factors such that when multiplied together they will result in an original polynomial. In the factorization method, we can reduce any algebraic or quadratic equation into its simpler form, where the equations are represented as the product of factors.
The equation we have, $3{{x}^{2}}+28x+32$
Multiplying the coefficient of \[{{x}^{2}}\] with the constant term, we get $3\times 32=96$
Now, we have to find two numbers whose sum is ‘28’ and multiplication is ‘96’.
Thus, the numbers are ‘24’ and ‘4’.
Hence, our equation can be written as
$\begin{align}
  & 3{{x}^{2}}+28x+32 \\
 & \Rightarrow 3{{x}^{2}}+24x+4x+32 \\
\end{align}$
Taking common ‘3x’ from first two terms and ‘4’ from last two terms, we get
$\Rightarrow 3x\left( x+8 \right)+4\left( x+8 \right)$
Again taking common $\left( x+8 \right)$ from both the terms, we get
$\Rightarrow \left( x+8 \right)\left( 3x+4 \right)$
This is the required solution of the given question.

Note: The above expression can also be factorized by completing the square method. First we have to reduce the coefficient of \[{{x}^{2}}\] by dividing the equation with the given coefficient of \[{{x}^{2}}\]. Then we have to express the whole equation in terms of subtraction of two square terms i.e. ${{a}^{2}}-{{b}^{2}}$. Then we can use the formula ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ to obtain the required factorization. This is the alternative method.