Question

Show that \[(x{\text{ }} - {\text{ }}2)\], \[(x{\text{ }} + {\text{ }}3)\] and \[(x{\text{ }} - {\text{ }}4)\] are the factors of \[{x^3}{\text{ }} - {\text{ }}{x^2}{\text{ }} - {\text{ }}10x{\text{ }} + {\text{ }}24\].

Answer 1

Hint:- Put \[x = 2, - 3\] and \[4\] in the given equation. We proceed with the factor theorem which states that if (x-a) is a factor of f(x) ,then f(a) =0.
Given \[(x{\text{ }} - {\text{ }}2)\], \[(x{\text{ }} + {\text{ }}3)\] and \[(x{\text{ }} - {\text{ }}4)\].

Complete step-by-step solution -
Let \[f(x)\] be a function of x.
\[ \Rightarrow \]And, \[f(x) = {x^3} - 3{x^2} - 10x + 24\] (1)
We know that,
\[ \Rightarrow \]If \[(x - a)\] is a factor of any function \[f(x)\].
Then, \[{\text{ }}f(a)\] should be 0.
So, if \[\left( {x - 2} \right)\] is a factor of \[f(x)\].
Then if we replace x with 2 in the equation . We get,
\[ \Rightarrow f(2) = ({(2)^3} - 3*{(2)^2} - 10*(2) + 24) = 8 - 12 - 20 + 24 = 0\]
As we have seen above, \[f(2) = 0\].
\[ \Rightarrow \]So, \[\left( {x - 2} \right)\] will be a factor of the given equation.
So, if \[\left( {x + 3} \right)\] is a factor of \[f(x)\].
Then if we replace x with -3 in equation 1. We get,
\[ \Rightarrow f( - 3) = ({( - 3)^3} - 3*{( - 3)^2} - 10*( - 3) + 24) = - 27 - 27 + 30 + 24 = 0\]
As we have seen above, \[f( - 3) = 0\].
\[ \Rightarrow \]So, \[\left( {x + 3} \right)\]will be a factor of the given equation.
So, if \[\left( {x - 4} \right)\] is a factor of \[f(x)\].
Then if we replace x with 4 in equation 1. We get,
\[ \Rightarrow f(4) = ({(4)^3} - 3*{(4)^2} - 10*(4) + 24) = 64 - 48 - 40 + 24 = 0\]
As we have seen above \[f(4) = 0\].
\[ \Rightarrow \]So, \[\left( {x - 4} \right)\] will also be a factor of the given equation.
\[ \Rightarrow \]Hence, \[(x{\text{ }} - {\text{ }}2)\], \[(x{\text{ }} + {\text{ }}3)\] and \[(x{\text{ }} - {\text{ }}4)\] are the factors of \[{x^3}{\text{ }} - {\text{ }}{x^2}{\text{ }} - {\text{ }}10x{\text{ }} + {\text{ }}24\].

Note:- Whenever we come up with this type of problem then the easiest and efficient way to check whether a number will be a factor of a given equation or not is by putting the number in the given equation. And if the value of the equation becomes zero, then that number is a factor of the given equation.