Question

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Answer 1

Answer:

( x + 1 )( x + 3 )( x - 14 )

Step-by-step explanation:

⇒ x^3 - 10x^2 - 53x - 42

     - 10x^2 = - 11x^2 + x^2

⇒ x^3 + x^2 - 11x^2 - 53x - 42

        - 53x =  - 11x - 42x

⇒ x^3 + x^2 - 11x^2 - 11x - 42x - 42

⇒ x^2( x + 1 ) - 11x( x + 1 ) - 42( x + 1 )

⇒ ( x + 1 )( x^2 - 11x - 42 )

⇒ ( x + 1 )( x^2 - 14x + 3x - 42 )

⇒ ( x + 1 )[ x( x - 14 ) + 3( x - 14 ) ]

⇒ ( x + 1 )[ ( x + 3 )( x - 14 ) ]

⇒ ( x + 1 )( x + 3 )( x - 14 )

   Hence,

 x^3 - 10x^2 - 53x - 42 = ( x + 1 )( x + 3 )( x - 14 )

Answer 2

QueStI0N -

Factorise -

$\sf{ x^3 - 10x^2 - 53x - 42 }$

Solution -

Let us first look at the given expression carefully .

$\sf{ x^3 - 10x^2 - 53x - 42 }$

Here, as we can observe , this is a Cubic polynomial .

Hence , it should be having three factors.

Now , Factorising cubic polynomials are a bit difficult and this is no difference .

Notice the given expression carefully again .

What are the patterns you can recognize in the polynomial ?

$\sf{ x^3 - 10x^2 - 53x - 42 }$

Here , as you have guesed correctly , the constant term here is 42 and the coefficent of x is 53 .

So, to have a simpler factor of x + 1, the coefficent of x should be 42 .

Then the remaining is 11x .

Now, this can be written as -

$\sf{ x^3 - 10x^2 - 11x - 42x - 42 }$

Now, seeing this expression this strikes -

-10x^2 can be written as x^2 - 11x^2

Now Substituting this into the above expression -

$\sf{ => x^3 + x^2 - 11x^2 - 11x - 42x - 42 } \\ \\ \sf{ => x^2 ( x + 1 ) - 11x ( x + 1 ) - 42 ( x + 1 ) } \\ \\ \sf{ => (x + 1 )( x^2 - 11x - 42 ) }$

Now, seeing the expression , x^2 - 11x - 42 , you may think that -

This can be written as -

$\sf{ => x^2 - 14x + 3x - 42 } \\ \\ \sf{ => x (x - 14 ) - 3 ( x - 14) } \\ \\ \sf{ => (x - 3)( x - 14 ) }$

Substituting this into the given expression into the original expression , we obtain the following result -

$\sf{ (x - 3)( x - 14)( x + 1 ) }$

Hence factorised