Question

2.2 Factorize the following by Factor theorem:
(a)x^3+9x^2+23x+15​

Answer 1

Answer is in the photo check it out....

HOPE IT WILL HELP ☺

#STAY HOME #STAY SAFE

Answer 2

Given :-

Cubic expression ;

p ( x ) = x³ + 9x² + 23x + 15

Required to find :-

• Factorised form of the given expression ?

Method used :-

• Performing long division by dividing the expression with one of its factor and Factorise the quotient

Concept used :-

• Factor theorem

Solution :-

Given :-

p ( x ) = x³ + 9x² + 23x + 15

We need to Factorise this above expression

So,

Let assume that ( x + 1 ) is the factor of p ( x )

Using factor theorem !

Let,

=> x + 1 = 0

=> x = - 1

Substitute this value in place of x in p ( x )

So,

p ( - 1 ) =

( - 1 )³ + 9 ( - 1 )² + 23 ( - 1 ) + 15 = 0

- 1 + 9 ( 1 ) - 23 + 15 = 0

- 1 + 9 - 23 + 15 = 0

- 24 + 24 = 0

- 24 & + 24 get cancelled due to opposite signs

0 = 0

LHS = RHS

Hence, our assumption is correct .

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

Now,

perform long division by dividing p ( x ) with ( x + 1 )

Since, x + 1 is the factor of p ( x )

So,

$\tt x \ + 1\big) {x}^{3} + 9 {x}^{2} + 23x + 15 \big( {x}^{2} + 8x + 15\\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt {x}^{3} + 1 {x}^{2} \\ \: \: \: \: \: \: \: \: \: \: \: \underline{ (-) \: (-) \: \: \: \: \: \: \: \: } \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \: \: \: + \: 8 {x}^{2} + 23x \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt + 8 {x}^{2} \: + \: \: \: 8x \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{ (-) \: \: \: \: \: ( - ) \: \: \: \: \: \: \: \: \: \: } \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt + 15x + 15 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt + 15x + 15 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{ ( - ) \: \: \: \: ( - ) \: \: \: \: \: \: \: \:} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \huge{ 0} \: \: \: \: \: }$

On performing long division ;

We get the quotient as x² + 8x + 15

Now,

We need to Factorise the quotient in order to find the other 2 factors

So,

$\rightarrowtail \rm \: {x}^{2} + 8x + 15 \: \\ \\ \rightarrowtail \rm \: {x}^{2} + 5x + 3x + 15 \: \\ \\ \rightarrowtail \rm \: x(x + 5) + 3(x + 5) \\ \\ \rightarrowtail \rm \: (x + 5)(x + 3)$

Hence,

$\tt {factors \: of \: p(x)} \begin{cases} \rm (x + 1) \\ \tt(x + 5) \\ \sf(x + 3) \end{cases}$

Therefore,

x³ + 9x² + 23x + 15 is factorised into 3 factors i.e. ( x + 1 ) ( x + 5 ) ( x + 3 )