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Find the Values of M and N So that X – 1 and X + 2 Both Are Factors of X3 + (3m + 1) X2 + Nx – 18. - ICSE Class 10 - Mathematics

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Question

Find the values of m and n so that x – 1 and x + 2 both are factors of x3 + (3m + 1) x2 + nx – 18.

Solution

Left `f(x)=x^3+(3m+1)x^2+nx-18` 

`x-1=0`⇒ x=1 

x-1 is a factor of f(x). so remainder =0 

∴ `(1)^3+(3m+1)(1)^2+n(1)-18=0` 

⇒`1+3m+1+n-18=0`  

⇒`3m+n-16=0`.......(1) 

x+2=0⇒ x=-2 

x+2 is a factor of f (x). so, remainder=0 

∴ `(-2)^3+(3m+1)(-2)^2+n(-2)-18=0`   

⇒`-8+12m+4-2n-18=0`

⇒`12m-2n-22=0` 

⇒`6m-n-11=0` ..........(2) 

Adding (1) and (2), we get, 

9m-27=0 

m=3 

putting the value of m in (1), we get, 

3(3)+n-16=0 

9+n-16=0  

n=7

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Solution Find the Values of M and N So that X – 1 and X + 2 Both Are Factors of X3 + (3m + 1) X2 + Nx – 18. Concept: Factor Theorem.
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