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\[{\text{ }}{x^3} - 5{x^2} - 2x + 24\]

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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.