Question

How do you simplify $\left( {x - 6} \right)\left( {x - 2} \right)\left( {x - 3} \right)$?

Answer 1

Hint: We are given a factored form of an algebraic expression and we have to find its expanded form in question. We are asked to simplify it. “Simplified” depends on what you are asking for. The factored form was what you gave in the problem. The expanded form is what you get by distributing out all terms; this is commonly referred to as the FOIL method. For this first we will take two factors and find the products and then by multiplying by third factors we will get the expanded form.

Complete step-by-step answer:
Step1: We are given an expression i.e.$\left( {x - 6} \right)\left( {x - 2} \right)\left( {x - 3} \right)$ this is the factored form of the expression and we have to find its expanded form. So we will use the FOIL method. Firstly we will expand $\left( {x - 6} \right)\left( {x - 2} \right)$ using the FOIL method. Now we will apply the distributive property of multiplication.
$ \Rightarrow \left\{ {x\left( {x + 3} \right) - 6\left( {x + 3} \right)} \right\}\left( {x - 5} \right)$
Again applying the distributive property.
$ \Rightarrow \left( {x.x + x.3 - 6x - 6.3} \right)\left( {x - 5} \right)$
Now we will simplify and combine the like terms. By first multiply$x$by$x$.
$ \Rightarrow \left( {{x^2} + x.3 - 6x - 6.3} \right)\left( {x - 5} \right)$
Moving$3$ to the left of$x$ then we will get:
$ \Rightarrow \left( {{x^2} + 3x - 6x - 6.3} \right)\left( {x - 5} \right)$
Multiply$ - 6$by$3$.
$ \Rightarrow \left( {{x^2} + 3x - 6x - 18} \right)\left( {x - 5} \right)$
Subtract$6x$from$3x$
$ \Rightarrow \left( {{x^2} - 3x - 18} \right)\left( {x - 5} \right)$
Step2: Expand $\left( {{x^2} - 3x - 18} \right)\left( {x - 5} \right)$ by multiplying each term in the first expression by each term in the second expression.
$ \Rightarrow {x^2}x + {x^2}.( - 5) - 3x.x - 3x.( - 5) - 18x - 18.( - 5)$
Multiply${x^2}$ b y$x$ by adding the exponents. To multiply${x^2}$ by $x$ raise $x$ to the power of$1$.
$ \Rightarrow {x^2}{x^1} + {x^2}.( - 5) - 3x.x - 3x.( - 5) - 18x - 18.( - 5)$
Use the power rule ${a^m}{a^n} = {a^{m + n}}$ to solve the exponents.
$ \Rightarrow {x^{2 + 1}} + {x^2}.( - 5) - 3x.x - 3x.( - 5) - 18x - 18.( - 5)$
$ \Rightarrow {x^3} + {x^2}.( - 5) - 3x.x - 3x.( - 5) - 18x - 18.( - 5)$
Add$2$and$1$. And also move$ - 5$to the left of${x^2}$
$ \Rightarrow {x^3} - 5.{x^2} - 3x.x - 3x.( - 5) - 18x - 18.( - 5)$
Now multiply $x$ by $x$ by adding the exponents. Move $x$ and also multiply $ - 5$ by $ - 3$ and $ - 18$ by $ - 5$.
$ \Rightarrow {x^3} - 5{x^2} - 3{x^2} + 15x - 18x + 90$
Now we will simplify by subtracting the $3{x^2}$ from $ - 5{x^2}$ and $18x$ from $15x$ we will get:
$ = {x^3} - 8{x^2} - 3x + 90$

Hence the expanded or simplified form is ${x^3} - 8{x^2} - 3x + 90$

Note:
When we are given a factored form and we have to find the expanded form. Then we will apply the FOIL method to find. This method is a simple multiplication method that can be applied on expansion of algebraic expression. Students mainly make mistakes while multiplying one bracket to another. Sometimes they make calculation mistakes due to confusion in multiplication of Signs. Just remember this if the minus sign is multiplied in the brackets then the sign gets reversed but if the plus is multiplied to the bracket then signs do not change. By keeping this concept in mind the question can be easily solved and never get wrong.