Question

factorize: x³– 3x²– 9x –5 urgent work please answer this question ​

Answer 1

Answer:

x³ - 3x² - 9x - 5

= x³ + x² - 4x² - 4x  - 5x - 5

= x²(x+1) - 4x(x+1) -5(x+1)

= (x+1)(x² - 4x - 5)

= (x+1)(x² - 5x + x - 5)

= (x+1)[x(x-5) + 1(x-5)]

= (x+1)[(x-5)(x+1)]

= (x+1)(x+1)(x-5)

=(x+1)²(x-5)

Answer 2

Given :-

Cubic polynomial :- x³ - 3x² - 9x - 5

Required to find :-

• Factorise the given cubic expression

Method used :-

• Factor theorem

• performing long division and Factorising the quotient

Solution :-

Given information :-

Cubic polynomial :- x³ - 3x² - 9x - 5

we need to Factorise the cubic expression .

let's consider this expression simply as p ( x ) .

Using Hit trial and error method ;

Let , x + 1 be the factor of p ( x ) .

So,

=> x + 1 = 0

=> x = - 1

p ( - 1 ) =

=> ( - 1 )³ - 3 ( - 1 )² - 9 ( - 1 ) - 5

=> - 1 - 3 ( 1 ) + 9 - 5

=> - 1 - 3 + 9 - 5

=> - 9 + 9

=> 0

Hence,

( x + 1 ) is the factor of p ( x ) .

Now,

Let's perform the long division ;

$\tt x + 1 \: \big) {x}^{3} - 3 {x}^{2} - 9x - 5 \big( {x}^{2} - 4x - 5\\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt {x}^{3} + \: \: \: {x}^{2} \\ \: \: \: \: \: \: \: \: \: \: \underline{ \: ( - ) \: ( - ) \: \: \: \: \: \: } \\ \: \: \: \: \: \: \: \: \: \: \tt \: \: \: \: \: \: \: \: \: \: \: \: - 4 {x}^{2} - 9x \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \: \: \: \: \: - 4x \: \: - 4x \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{ \: ( + ) \: \: \: \: ( + ) \: \: \: \: \: \: } \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: - 5x - 5 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \: - 5x - 5 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{ ( + ) \: \: ( + ) \: \: \: } \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{ \: \: \: \: \: \tt \: \: \: \: \: \: 0 \: \: \: \: \: \: \: }$

Hence,

Quotient = x² - 4x - 5

Now,

we need to Factorise this quotient in order to find the other factors of p ( x )

So,

➜ x² - 4x - 5

➜ x² - 5x + 1x - 5

➜ x ( x - 5 ) + 1 ( x - 5 )

➜ ( x - 5 ) ( x + 1 )

Therefore,

x³ - 3x² - 9x - 5 can be factorised into ( x + 1 ) , ( x - 5 ) & ( x + 1 )

This implies,

x³ - 3x² - 9x - 5 = ( x + 1 )² & ( x - 5 )