A third-degree polynomial P(x)=x3+2x2+mx+n divided by x2−4x+3 has a remainder of 3x+2. Then the value of n = ....
Explanation
The basic concept of polynomial division is
P(x)=g(x)⋅h(x)+s(x)
where P(x) is the polynomial being divided, g(x) is the divisor, h(x) is the quotient, and s(x) is the remainder.
Thus
x3+2x2+mx+n=(x2−4x+3)h(x)+(3x+2)
x3+2x2+mx+n=(x−1)(x−3)h(x)+(3x+2)
where P(x)=x3+2x2+mx+n, g(x)=(x−1)(x−3), and s(x)=3x+2.
Substitute x=1
P(1)=(1−1)(1−3)h(1)+(3(1)+2)
13+2(1)2+m(1)+n=0+5
1+2+m+n=5
m+n=2…(1)
Substitute x=3
P(3)=(3−1)(3−3)h(3)+(3(3)+2)
33+2(3)2+m(3)+n=0+11
27+18+3m+n=11
45+3m+n=11
3m+n=−34…(2)
Eliminate both equations
3m+nm+n(2)−(1):(3m+n)−(m+n)−2mm=−34…(2)=2…(1)=−34−2=36=−18
Substitute into equation (1)
m+n=2
−18+n=2
n=20