Answer
Verified
481.5k+ views
Hint: Put the different values of $x$ in the given equation until the value of the equation becomes zero.
We have to find such value which will make the whole expression equal to zero
Such that
$P(x) = {x^3} - 5{x^2} - 2x + 24$
Let $x = 1$, we get,
$P(1) = {(1)^3} - 5{(1)^2} - 2(1) + 24$
$P(1) = 1 - 5 - 2 + 24 = 13 \ne 0$
Now, let ${\text{ }}x = 2$
$P(2) = {(2)^3} - 5{(2)^2} - 2(2) + 24$
$P(2) = 8 - 20 - 4 + 24 = 8 \ne 0$
Now, let${\text{ }}x = - 2$
$P( - 2) = {( - 2)^3} - 5{( - 2)^2} - 2( - 2) + 24$
$P( - 2) = - 8 - 20 + 4 + 24 = 0$
Since the value comes out to be zero,
Therefore, \[x + 2\]is one of the factors.
The other factors can be calculated by dividing the given expression
${x^3} - 5{x^2} - 2x + 24{\text{ by }}x + 2.$
That is,
${\text{(}}{x^3} - 5{x^2} - 2x + 24) \div {\text{(}}x + 2)$
$\begin{gathered}
{\text{ }}{x^2} - 7x + 12 \\
x + 2\left){\vphantom{1{{x^3} - 5{x^2} - 2x + 24}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{{x^3} - 5{x^2} - 2x + 24}}} \\
{\text{ }}{x^3} + 2{x^2} \\
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\
{\text{ - 7}}{x^2} - 2x + 24 \\
{\text{ - 7}}{x^2} - 14x \\
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\
{\text{ 1}}2x + 24 \\
{\text{ 1}}2x + 24 \\
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\
{\text{ 0}} \\
\end{gathered} $
Thus, we get, the other factor as ${x^2} - 7x + 12$
Which can be written as
\[ = {x^2} - 4x - 3x + 12\]
$ = x(x - 4) - 3(x - 4)$
$ = (x - 3)(x - 4)$
Hence, the required solution: ${\text{(}}{x^3} - 5{x^2} - 2x + 24) = (x + 2)(x - 3)(x - 4)$
Note: The first factor must be chosen very carefully as the other factors are determined on its basis only. Later we have the divide the cubic equation by the first factor to convert it into a quadratic equation and factorised to find the remaining factors.
We have to find such value which will make the whole expression equal to zero
Such that
$P(x) = {x^3} - 5{x^2} - 2x + 24$
Let $x = 1$, we get,
$P(1) = {(1)^3} - 5{(1)^2} - 2(1) + 24$
$P(1) = 1 - 5 - 2 + 24 = 13 \ne 0$
Now, let ${\text{ }}x = 2$
$P(2) = {(2)^3} - 5{(2)^2} - 2(2) + 24$
$P(2) = 8 - 20 - 4 + 24 = 8 \ne 0$
Now, let${\text{ }}x = - 2$
$P( - 2) = {( - 2)^3} - 5{( - 2)^2} - 2( - 2) + 24$
$P( - 2) = - 8 - 20 + 4 + 24 = 0$
Since the value comes out to be zero,
Therefore, \[x + 2\]is one of the factors.
The other factors can be calculated by dividing the given expression
${x^3} - 5{x^2} - 2x + 24{\text{ by }}x + 2.$
That is,
${\text{(}}{x^3} - 5{x^2} - 2x + 24) \div {\text{(}}x + 2)$
$\begin{gathered}
{\text{ }}{x^2} - 7x + 12 \\
x + 2\left){\vphantom{1{{x^3} - 5{x^2} - 2x + 24}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{{x^3} - 5{x^2} - 2x + 24}}} \\
{\text{ }}{x^3} + 2{x^2} \\
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\
{\text{ - 7}}{x^2} - 2x + 24 \\
{\text{ - 7}}{x^2} - 14x \\
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\
{\text{ 1}}2x + 24 \\
{\text{ 1}}2x + 24 \\
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\
{\text{ 0}} \\
\end{gathered} $
Thus, we get, the other factor as ${x^2} - 7x + 12$
Which can be written as
\[ = {x^2} - 4x - 3x + 12\]
$ = x(x - 4) - 3(x - 4)$
$ = (x - 3)(x - 4)$
Hence, the required solution: ${\text{(}}{x^3} - 5{x^2} - 2x + 24) = (x + 2)(x - 3)(x - 4)$
Note: The first factor must be chosen very carefully as the other factors are determined on its basis only. Later we have the divide the cubic equation by the first factor to convert it into a quadratic equation and factorised to find the remaining factors.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Change the following sentences into negative and interrogative class 10 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE