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# If ax 3 +3x 2 +bx−3 has a factor (2x+3) and leaves remainder −3 when divided by (x+2), find the values of a and b. With these values of a and b, factorise the given expression.

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A basic question from factorisation chapter in which we have been asked to find the values of and b If ax3+3x2+bx3 has a factor (2x+3) and leaves remainder 3 when divided by (x+2),  With these values of a and b, factorise the given expression.

ML Aggarwal(avichal publication), Factorisation, chapter 6, question no 26

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1. Let
p(x)=ax3+3x2+bx3
g(x)=2x+3=0x=23
f(x)=x+2=0x=2

Given
g(x) is a factor of f(x)

By factor theorem,
p(23)=0
a(23)3+3(23)2+b(23)3=0
a(827)+3(49)+b(23)3=0
827a+42723b3=0
8(−27a+5412b)=3
27a+5412b=24
3(9a+4b)=2454=30
9a+4b=10...(i)

Also, p(x) when divided by f(x) leaves a remainder 3

By remainder theorem,
p(2)=3
a(2)3+3(2)2+b(2)3=3
8a+122b=0
8a+2b=12
4a+b=6...(ii)

Solving (i)  and  (ii), we get
a=2  and  b=2

Hence p(x)=2x3+3x22x3
=x2(2x+3)(2x+3)==(2x+3)(x21)
=(2x+3)(x+1)(x1)

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