If P(x) = x^3 + ax^2 + 9x + b , find out a and b real numbers knowing that P(x) is divisible by x^2 - 4x + 5 .
P(x)= x^3+ ax^2 +9x+b
P(x)is divided by x^2-4x+5
==> P(x)= (x^2-4x +5) Q(x)
= (x+1)(x-5) Q(x)
Then x1=-1 and x1=5 are solutions fpr P(x)
P(-1)= -1 +a -9 +b= 0
= a+b -10=0 ....(1)
P(5)= 125 + 25a +45 +b =0
= 25a +b +170 =0 ....(2)
subtract (1) from (2)
==> 24a +180 =0
==> a = -180/24 = -7.5
==> b = 10 -a = 10+7.5 = 17.5
Then P(x)= x^3 -7.5x^2+9x+17.5
Because of the fact that P(x) is a 3rd degree polynomial and it's divisible by x^2 - 4x +5, we could write P(x) as multiplication between x^2 - 4x +5 an unknown first degree polynomial, cx + d.
P(x) = (x^2 - 4x +5)(cx + d)
P(x) is similar with the expression from the right side of the equal, if only corresponding quotients from the both sides of the equal are the same.
P(x) = cx^3 + dx^2 - 4cx^2 - 4dx + 5cx + 5d
x^3 + ax^2 + 9x + b = cx^3 + x^2*(d - 4c) + x*(5c - 4d) + 5d
c = 1
d - 4c = a, d - 4 = a
5c - 4d = 9, 5 - 4d = 9, - 4d = - 5 + 9, -4d = 4, d = -1
d-4 = a, d = -1, -1- 4 = a, a = - 5
b = 5d, d = - 1, b = - 5
f(x) = x^3+ax^2+9x+b. Given (x) is divisible by x^2-4x+5.To determine a and b.
The dividing expression, x^2-4x+5 , has the factors (x-5) and x+1.
Therefore P(x) = x^3+ax^2+9x+ b = (x-5)(x+1)(x+k) say
Then the x's on both sodes should be equal. So ,
9x = x(k-5k-5) Or
9= -4k-5. Or 4k = -14 . Or k = -14/4 = -7/2
So the third root is x = 3.5
Therfore a = -sumof roots = -(5-1+3.5 ) = - 7.5
b = - product of all three roots = -5*(-1)(3.5) = 17.5.