When f(x) = x^3 + 3x^2 + ax + b is divided by x + 2 and x - 1, the remainders are -4 and 8, respectively. Find the values of a and b.
+129840 When f(x) = x^3 + 3x^2 + ax + b is divided by x + 2 and x - 1, the remainders are -4 and 8, respectively. Find the values of a and b.
By the Remainder Theorem, f(-2) = -4 and f(1) = 8........so.....we can set up this system :
[-2]^3 + 3[-2]^2 + a(-2) + b = -4
[1]^3 + 3[1]^2 + a (1) + b = 8 simplify these
-8 + 12 - 2a + b = -4
1 + 3 + a + b = 8 simplify again
-2a + b = -8
a + b = 4 (1)
Multiply (1) through by - 1
-2a + b = -8
-a - b = -4 add the equations
-3a = -12 → a = 4
And.......using (1) to find b
4 + b = 4 → b = 0
So.....the original function is
f(x) = x^3 + 3x^2 + 4x
Here's the graph : https://www.desmos.com/calculator/xi4bdnrstr
