Question

How do you simplify $\left( 4x-1 \right)\left( 2x-1 \right)\left( 3x-2 \right)$ ?

Answer 1

Hint: We can solve problems by simplifying expressions like this using the two methods, such as the distributive method and the FOIL method. We first simplify the first two terms in the brackets using the FOIL method, according to which all the entities in the bracket will be multiplied to all the entities of the next bracket. After that we apply the distribution method between the terms we got after applying the FOIL method and the terms in the last bracket. Further simplifying we get the result.

Complete step by step solution:
The expression we have is
$\left( 4x-1 \right)\left( 2x-1 \right)\left( 3x-2 \right)$
We now apply the FOIL method for the entities in the first two brackets. According to this method all the entities in the bracket will be multiplied with all the entities in the next bracket as shown below
$\left( a+b \right)\left( c+d \right)=ac+ad+bc+bd$
Hence, from the given expression we write
$=\left( 4x\cdot 2x-1\cdot 2x-4x\cdot 1+1\cdot 1 \right)\left( 3x-2 \right)$
Further completing the above calculations, we get
$=\left( 8{{x}^{2}}-2x-4x+1 \right)\left( 3x-2 \right)$
Further adding the like terms in the above expression, we get
$=\left( 8{{x}^{2}}-6x+1 \right)\left( 3x-2 \right)$
Now, we apply the distribution method between the term of the two brackets as shown below
$=\left( 8{{x}^{2}}\cdot 3x-6x\cdot 3x+1\cdot 3x-8{{x}^{2}}\cdot 2+6x\cdot 2-1\cdot 2 \right)$
Further completing the above calculations, we get
$=\left( 24{{x}^{3}}-18{{x}^{2}}+3x-16{{x}^{2}}+12x-2 \right)$
Now, we add the like terms in the above expression and write it in a manner of ascending power of $x$ as shown below
$=24{{x}^{3}}-34{{x}^{2}}+15x-2$
Therefore, we conclude that the simplified form of the given expression is $24{{x}^{3}}-34{{x}^{2}}+15x-2$ .

Note:
While multiplying terms applying methods such as the FOIL method or the distribution method, we must be careful about the signs of the entities in the bracket to get the correct result. Also, we must not perform step jumps specifically for problems like this, as it may cause complications and errors in the problem.