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...
Answers
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