Let f(x) = ax^3 + bx^2 + 3x – 4. If f(x) is divided by (x – 2), the remainder is 2. When it is divided by (x – 1), the remainder is -2. Find a and b.
testing
To find the values of a and b in the equation f(x) = ax^3 + bx^2 + 3x - 4, we can use the information provided about the remainders when f(x) is divided by (x - 2) and (x - 1).
When f(x) is divided by (x - 2), the remainder is 2. This means that if we substitute x = 2 into the equation, we should get a remainder of 2. Let's do that:
f(2) = a(2)^3 + b(2)^2 + 3(2) - 4
= 8a + 4b + 6 - 4
= 8a + 4b + 2
Since the remainder is 2, we can set this expression equal to 2:
8a + 4b + 2 = 2
Simplifying this equation, we find:
8a + 4b = 0 (Equation 1)
Now let's consider the second piece of information. When f(x) is divided by (x - 1), the remainder is -2. Substituting x = 1 into the equation, we should get a remainder of -2:
f(1) = a(1)^3 + b(1)^2 + 3(1) - 4
= a + b + 3 - 4
= a + b - 1
Setting this expression equal to -2, we have:
a + b - 1 = -2
Simplifying this equation, we find:
a + b = -1 (Equation 2)
Now we have a system of two equations (Equation 1 and Equation 2) with two unknowns (a and b). We can solve this system of equations to find the values of a and b.
Multiplying Equation 1 by 2, we get:
16a + 8b = 0 (Equation 3)
Subtracting Equation 2 from Equation 3, we eliminate b:
16a + 8b - (a + b) = 0 - (-1)
15a + 7b = 1
Now we can solve this equation for a:
15a + 7b = 1
15a = 1 - 7b
a = (1 - 7b) / 15
Substituting this expression for a back into Equation 2, we can solve for b:
a + b = -1
(1 - 7b) / 15 + b = -1
1 - 7b + 15b = -15
8b = -16
b = -2
Finally, substituting b = -2 back into the expression for a:
a = (1 - 7b) / 15
a = (1 - 7(-2)) / 15
a = (1 + 14) / 15
a = 15 / 15
a = 1
Therefore, the values of a and b are a = 1 and b = -2.