If x - 1 and X + 2 a factor of x^3+ 10x^2 + ax + b, then find the value of a and b
Answers
Answered by
3
By factor theorem, if (x - a) is a factor of a polynomial P(x) in x, then P(a) = reminder of P(x) when divided by (x - a) = 0.
P(1) = 1^3 + 10*1^2 + p * 1 + q = 0 \\ \\ 1 + 10 + p + q = 0 \\ \\ p + q = -11 \\ \\ \\ P(-2) = -2^3 + 10*(-2)^2 + p(-2) + q = 0 \\ \\ -8 + 40 -2 p + q = 0 \\ \\ -2 p + q = -32 \\ \\ 2p - q = 32 \\ \\ Add\ the\ two\ equations, \\ \\ p+q+2p -q = -11+32 \\ \\ 3p = 21 \\ \\ p = 7 \\ \\ substitute\ this\ in\ p+q = -11,\ to\ get\ \\ \\ q = -18 \\
P(1) = 1^3 + 10*1^2 + p * 1 + q = 0 \\ \\ 1 + 10 + p + q = 0 \\ \\ p + q = -11 \\ \\ \\ P(-2) = -2^3 + 10*(-2)^2 + p(-2) + q = 0 \\ \\ -8 + 40 -2 p + q = 0 \\ \\ -2 p + q = -32 \\ \\ 2p - q = 32 \\ \\ Add\ the\ two\ equations, \\ \\ p+q+2p -q = -11+32 \\ \\ 3p = 21 \\ \\ p = 7 \\ \\ substitute\ this\ in\ p+q = -11,\ to\ get\ \\ \\ q = -18 \\
Similar questions