Question

How do you factor $6{{b}^{3}}-4{{b}^{2}}?$

Answer 1

Hint: This question shows a polynomial with the highest degree of 2. A polynomial is nothing but an expression consisting of coefficients and variables separated by operators such as addition, subtraction, etc. We can solve this polynomial by factoring the terms and this can be found out by using the Greatest Common Factor method.

Complete step by step answer:
In the given question, we have $6{{b}^{3}}-4{{b}^{2}}$ as the equation to be factorized. To factorize this expression, we use the following method.
Looking at the equation, we can see that there is no constant term and that there are some terms common in both the terms of the expression. We shall factorize this equation by taking out those common terms from both the terms and for that we shall calculate the Greatest Common Factor for each of the terms.
$\Rightarrow 6{{b}^{3}}-4{{b}^{2}}$
Looking at the equation, we can take the number out common for both the terms, and to find this number we perform the Greatest Common Factor (GCD) method. This method states that it will select a term common such that it is the greatest number that divides both the terms.
For the above equation,
$\Rightarrow GCD\left( 6,4 \right)=2$
So, 2 is the greatest number that divides both 6 and 4. Taking this number out common,
$\Rightarrow 2\left( 3{{b}^{3}}-2{{b}^{2}} \right)$
Similarly, we check for the common term and compute the GCD for that.
$\Rightarrow GCD\left( {{b}^{3}},{{b}^{2}} \right)={{b}^{2}}$
So, ${{b}^{2}}$ is the greatest term that divides both ${{b}^{3}}$ and ${{b}^{2}}.$
Now we can take a ${{b}^{2}}$ term out common too,
$\Rightarrow 2{{b}^{2}}\left( 3b-2 \right)$

Hence the final expression after factorization is $2{{b}^{2}}\left( 3b-2 \right).$

Note: While solving this question, the students need to be careful while performing the GCD and taking the terms out. If there are more than 2 terms, then the term to be taken out should be taken out common from every term. We can also solve this by taking the GCD for the terms as a whole instead of splitting it as a number part and variable part. But again, this method is a little more complex and care should be taken if this method is used.