Question

How do you factor $3{{x}^{2}}-12$?

Answer 1

Hint: Find the common constant factor of both the terms. Then take that common factor out first. Convert the remaining factor into ${{a}^{2}}-{{b}^{2}}$ form. Then use the formula ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ for further factorization.

Complete step by step solution:
Factorization: A polynomial can be written as a product of two or more polynomials of degree less than or equal to that of it. Each polynomial involved in the product will be a factor of it. The process involved in breaking a polynomial into the product of its factors is known as the factorization of polynomials.
The expression we have $3{{x}^{2}}-12$,
As we know both the terms are divisible by ‘3’
So, the constant common factor of both the terms is ‘3’
Taking the common factor ‘3’ out from the given expression, we get
$\Rightarrow 3\left( {{x}^{2}}-4 \right)$
For $\left( {{x}^{2}}-4 \right)$ part,
‘4’ can be written as ${{\left( 2 \right)}^{2}}$
So, $\left( {{x}^{2}}-4 \right)$can be written as ${{\left( x \right)}^{2}}-{{\left( 2 \right)}^{2}}$
As we know, ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$
So, ${{\left( x \right)}^{2}}-{{\left( 2 \right)}^{2}}=\left( x+2 \right)\left( x-2 \right)$
Hence, our expression becomes
$\Rightarrow 3\left( x+2 \right)\left( x-2 \right)$
This is the required factorization of the given question.

Note: The common constant factor should be taken out first as it is a factor itself. The solution should be in maximum simplified form. For example, the factor $3\left( {{x}^{2}}-4 \right)$ we obtained during the calculation can be the solution. But for appropriate solution it should be factored using the formula ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ to get the final result as $3\left( x+2 \right)\left( x-2 \right)$.