Question

if (x+2) and (x-1) are factors of x³+10x²+m+n then find m and n


plz answer this...plz answer if u know the right answer...​

Answer 1

Step-by-step explanation:

have

p(1) = 0

We have p(x)=x3+10x2+mx+np(x)=x3+10x2+mx+n

Substituting x= 1 in the expression of p(x), we get

p(1)=(1)3+10(1)2+m(1)+n=1+10+m+n=m+n+11p(1)=(1)3+10(1)2+m(1)+n=1+10+m+n=m+n+11

But since p(1) = 0, we have

m+n+11=0 (ii)m+n+11=0 (ii)

Multiplying equation (ii) by 2 and adding equation (i) and (ii), we get

−2m+2m+n+2n+32+22=0⇒3n+54=0−2m+2m+n+2n+32+22=0⇒3n+54=0

Subtracting 54 from both sides, we get

3n=−543n=−54

Dividing by 3 on both sides, we get

n=−543=−18n=−543=−18

Substituting the value of n in equation (ii), we get

m−18+11=0⇒m−7=0m−18+11=0⇒m−7=0

Adding 7 on both sides, we get

m=7m=7

Hence, we have

m=7 and n = -18

Hence option cc is correct.

Note: Alternative solution:

We will use the property that if α,β,γα,β,γ are the roots of the polynomial p(x)=ax3+bx2+cx+dp(x)=ax3+bx2+cx+d, then we have

α+β+γ=−ba,αβ+βγ+γα=ca,αβγ=−daα+β+γ=−ba,αβ+βγ+γα=ca,αβγ=−da

Let the third root of the polynomial p(x) be a.

Hence, we have

a+1+(−2)=−10⇒a=−9a+1+(−2)=−10⇒a=−9

Hence the third root is -9

Now, we have

m=(1)(−2)+(−9)(−2)+(1)(−9)=−2+18−9=7m=(1)(−2)+(−9)(−2)+(1)(−9)=−2+18−9=7

Also, we have

n=−(1)(−2)(−9)=−18