Question

♠️ Class 9
♣ Chapter - polynomials

Question - Factorise : x³ + 13 x² + 32x + 20
♪ Brainly Moderators
♪ Brainly Stars​

Answer 1

In the question we have to Factorize x³ + 13 x² + 32x + 20

So now the factors of + 20 are -

+1 , -1 , +2 , -2 , +4 , -4 , +5 , -5 , +10 , -10 , +20 , -20

By hit and trial we find that

• f(-10)=0 , So x+10 is the factors of p(x)

${\sf{:\implies f( x) \: = {x}^{3} + 13 {x}^{2} - 32x + 20 }}$

${\sf{:\implies f( - 10) \: = - 1000 + 1300 - 320 + 20 }}$

${\sf{:\implies 300 - 320 + 20 }}$

${\sf{:\implies - 20 + 20 }}$

${\sf{:\implies 0 }}$

Hence, the 1st factor is ( x+10 )

Now we know theat (x+10) is the factor of x³ + 13 x² + 32x + 20 and if we will divide x+10 by x³ + 13 x² + 32x + 20 we will find the another factor.

On Dividing -

$\begin{gathered} \tt \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: {x}^{2} + 3x + 2 \\ \tt x + 10\overline{| \cancel{{x}^{3}} + {13x}^{2} + 32x + 20 } \\ \tt \: \: \: \: \: \: \: \underline{ - \cancel{{x}^{3}} + {10x}^{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: } \\ \tt \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \cancel{3 {x}^{2} }+ 32x + 20 \\ \tt \: \: \: \: \: \: \: \underline{- \: \: \: \: \: \: \: \: \cancel{3x ^{2} }+ 30x \: \: \: }\\ \tt \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \cancel{ 2x + 20} \\ \tt \: \: \: \: \ \: \: \: \underline{ - \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \cancel{2x+ 20x } \: \: \: }\\ \tt \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 0 \end{gathered}$

Hence, the 2st factor is x² + 3x + 2

Now we will factorise x² + 3x + 2

${\sf{:\implies {x}^{2} + 3x + 2 }}$

${\sf{:\implies {x}^{2} + 2x + 1x+ 2 }}$

${\sf{:\implies {x}(x + 2)+ 1(x + 2)}}$

${\sf{:\implies (x + 1)(x + 2)}}$

Hence , the factors of x³ + 13 x² + 32x + 20 are (x+10) (x+1) (x+2) .