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(3x + 5) is a Factor of the Polynomial (A – 1)X3 + (A + 1)X2 – (2a + 1)X – 15. Find the Value of ‘A’, Factorise the Given Polynomial Completely. - ICSE Class 10 - Mathematics

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Question

(3x + 5) is a factor of the polynomial (a – 1)x3 + (a + 1)x2 – (2a + 1)x – 15. Find the value of ‘a’, factorise the given polynomial completely. 

Solution

Let f(x)=(a-1)x^3+(a-1)x-15 

it is given that (3x+5) is a factor of f(x) 

∴ Remainder =0 

`f((-5)/3)=2` 

`(a-1)(-5/3)^3+(a+1)((-5)/3)^2-(2a+1)((-5)/3)-15=0`

 `(a-1)((-125)/27)+(a+1)(a+1)(25/9)-(2a+1)((-5)/3)-15=0` 

`(-125(a-1)+75(a+1)+45(2a+1)-405)/27=0` 

`-125a+125+75a+75+90a+45-405=0` 

`40a-160=0` 

`40a=160` 

`a=4` 

∴ `f(x)=(a-1)x^3+(a+1)x^2-(2a+1)x-15` 

`=3x^3+5x-9x-15 `

 

∴ `3x^3+5x^2-9x-15=(3x+5)(x^2-3)` 

                          =`(3x+5)(x+sqrt3)(x-sqrt3)`

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Solution (3x + 5) is a Factor of the Polynomial (A – 1)X3 + (A + 1)X2 – (2a + 1)X – 15. Find the Value of ‘A’, Factorise the Given Polynomial Completely. Concept: Factor Theorem.
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