Question

Simplify the following:
\[(i)\]\[(x + 6)(x + 4)(x - 2)\]
\[(ii)\]\[(x - 6)(x - 4)(x + 2)\]
\[(iii)\]\[(x + 6)(x - 4)(x - 2)\]
\[(iv)\]\[(x - 6)(x + 4)(x - 2)\]

Answer 1

Hint: We are provided a set of questions and we have to simplify them one by one.
In order to solve this question, we are going to use identity \[\left( {x + a} \right)\left( {x + b} \right)\left( {x + c} \right) = {x^3} + \left( {a + b + c} \right){x^2} + \left( {ab + bc + ca} \right)x + abc\]
Therefore, \[\left( {x + a} \right)\left( {x + b} \right)\left( {x + c} \right) = \]\[{x^3}\]+ (Sum of the constant term) \[{x^2}\]+ (Sum of the constant term taken two at a time) \[x + \] product of constant terms.
We can also simplify using simple distribution and algebraic properties. We will multiply the first two terms first and then with the third term. We can use any method.

Complete answer:
We will use the above identity to simplify these question

\[(i)\]\[(x + 6)(x + 4)(x - 2)\]
Using identity:
\[\left( {x + a} \right)\left( {x + b} \right)\left( {x + c} \right) = {x^3} + \left( {a + b + c} \right){x^2} + \left( {ab + bc + ca} \right)x + abc\]
Here, \[a = 6\] \[b = 4\]and \[c = - 2\]
Substituting the value of \[a\]\[,\]\[b\] and \[c\]in the above identity we get
=\[(x + 6)(x + 4)(x - 2)\]
\[ = \] \[{x^3} + \left( {6 + 4 - 2} \right){x^2} + \left( {6 \times 4 + 4 \times - 2 + - 2 \times 6} \right) + 6 \times 4 \times - 2\]
\[ = \] \[{x^3} + 8{x^2} + 4x - 48\]

\[(ii)\]\[(x - 6)(x - 4)(x + 2)\]
Using identity:
 \[\left( {x + a} \right)\left( {x + b} \right)\left( {x + c} \right) = {x^3} + \left( {a + b + c} \right){x^2} + \left( {ab + bc + ca} \right)x + abc\]
Here, \[a = - 6\] \[b = - 4\]and \[c = 2\]
Substituting the value of \[a\]\[,\]\[b\] and \[c\]in the above identity we get
\[(x - 6)(x - 4)(x + 2)\]\[ = \]\[{x^3} - 8{x^2} + 4x + 48\]

\[(iii)\]\[(x + 6)(x - 4)(x - 2)\]
Using identity:
\[\left( {x + a} \right)\left( {x + b} \right)\left( {x + c} \right) = {x^3} + \left( {a + b + c} \right){x^2} + \left( {ab + bc + ca} \right)x + abc\]
Here, \[a = 6\] \[b = - 4\] and \[c = - 2\]
Substituting the value of \[a\]\[,\]\[b\] and \[c\]in the above identity we get
\[(x + 6)(x - 4)(x - 2)\]\[ = \] \[{x^3} + \left( {6 - 4 - 2} \right){x^2} + \left( {6 \times - 4 + - 4 \times - 2 + - 2 \times 6} \right)x + 6 \times - 4 \times - 2\]
\[ = \] \[{x^3} + \left( {6 - 4 - 2} \right){x^2} + \left( {6 \times - 4 + - 4 \times - 2 + - 2 \times 6} \right)x + 6 \times - 4 \times - 2\]
\[ = \] \[{x^3} - 28x + 48\]

\[(iv)\]\[(x - 6)(x + 4)(x - 2)\]
Using identity:
\[\left( {x + a} \right)\left( {x + b} \right)\left( {x + c} \right) = {x^3} + \left( {a + b + c} \right){x^2} + \left( {ab + bc + ca} \right)x + abc\]
Here, \[a = - 6\] \[b = 4\]and \[c = - 2\]
Substituting the value of \[a\]\[,\]\[b\] and \[c\]in the above identity we get
\[(x - 6)(x + 4)(x - 2)\]\[ = \] \[{x^3} + \left( { - 6 + 4 - 2} \right){x^2} + \left( { - 6 \times 4 + 4 \times - 2 + - 2 \times - 6} \right)x + - 6 \times 4 \times - 2\]
\[ = \] \[{x^3} - 4{x^2} - 20x + 48\]

Note: Avoid calculation mistakes because they can lead to an incorrect solution. We can cross-verify it by opening parenthesis and solving.
The student must note that there is an alternate way to solve the same question
Alternate way:
We will use simple distribution and algebraic properties to solve this question.
\[(i)\] \[(x + 6)(x + 4)(x - 2)\]
\[ = \] Firstly, multiply the first two terms we get
\[\left( {x + 6} \right)\left( {x + 4} \right) = {x^2} + 10x + 24\]
Now we will multiply it with the third term
\[x\left( {{x^2} + 10x + 24} \right) - 2\left( {{x^2} + 10x + 24} \right)\]
\[ = {x^3} + 10{x^2} + 24x - 2{x^2} - 20x + 48\]
Now combining the like term, we get
\[{x^3} + 8{x^2} + 4x + 48\]
\[(ii)\]\[(x - 6)(x - 4)(x + 2)\]
Firstly, multiply the first two terms we get
\[\left( {x - 6} \right)\left( {x - 4} \right) = {x^2} - 10x + 24\]
Now we will multiply it with the third term
\[x\left( {{x^2} - 10x + 24} \right) + 2({x^2} - 10x + 24)\]
\[ = {x^3} - 10{x^2} + 24x + 2{x^2} - 20x + 48\]
Now combining the like term, we get
\[ = \] \[{x^3} - 8{x^2} + 4x + 48\]
\[(iii)\] \[(x + 6)(x - 4)(x - 2)\]
Firstly, multiply the first two terms we get
\[\left( {x + 6} \right)\left( {x - 4} \right) = {x^2} + 2x - 24\]
Now we will multiply it with the third term
\[x\left( {{x^2} + 2x - 24} \right) - 2\left( {{x^2} + 2x - 24} \right)\]
\[ = {x^3} + 2{x^2} - 24x - 2{x^2} - 4x + 48\]
Now combining the like term, we get
\[{x^3} - 28x + 48\]
\[(iv)\] \[(x - 6)(x + 4)(x - 2)\]
Firstly, multiply the first two terms we get
\[\left( {x - 6} \right)\left( {x + 4} \right) = {x^2} - 2x - 24\]
Now we will multiply it with the third term
\[x\left( {{x^2} - 2x - 24} \right) - 2\left( {{x^2} - 2x + 24} \right)\]
\[{x^3} - 2{x^2} - 24x - 2{x^2} + 4x - 48\]
Now combining the like term, we get
\[{x^3} - 4{x^2} - 20x + 48\]