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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

#### 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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