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# Use factor theorem to factorize the following polynomials completely. x 3 +2x 2 -5x -6

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We have been asked to find to factorize the following polynomials completely, using factor theorem in the given polynomial
x3+2x25x6

Book – ML Aggarwal, Avichal Publication, factorisation, chapter 6, question no 15(i)

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1. To find a factor ofÂ f(x)Â we assume different values ofÂ xÂ and substitute it inÂ f(x)

First we consider the factors of the constant termÂ âˆ’6Â i.e.,Â Â±1,Â±2,Â±3,Â±6
Now we substitute the values ofÂ xÂ inÂ f(x)
IfÂ f(x)=0Â for some valueÂ a
ThenÂ (xâˆ’a)Â is a factor ofÂ f(x)
[Note : This is a hit and trial method, So you have to iterate the process until you get the requiredÂ x]

SubstitutingÂ Â x=âˆ’1Â inÂ f(x)Â , We get,
f(âˆ’1)=(âˆ’1)3+2(âˆ’1)2âˆ’5(âˆ’1)âˆ’6
=âˆ’1+2(1)+5âˆ’6
=âˆ’1+2+5âˆ’6
=âˆ’7+7
=0
âˆ´Â By factor theorem,
(x+1)Â is a factor ofÂ f(x)

Now dividingÂ f(x)Â byÂ (x+1), we get
x3+2x2âˆ’5xâˆ’6=(x+1)(x2+3xâˆ’2xâˆ’6)=(x+1){x(x+3)âˆ’2(x+3)}=(x+1)(xâˆ’2)(x+3)

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