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.
To find the values of a and b, we need to use polynomial division and the remainder theorem.
First, let's divide f(x) by (x - 2) and see what the remainder is.
We know that when f(x) is divided by (x - 2), the remainder is 2. This means that f(2) = 2.
To find f(2), substitute x = 2 into f(x) and solve for it:
f(2) = a(2)^3 + b(2)^2 + 3(2) - 4
= 8a + 4b + 6 - 4
= 8a + 4b + 2.
Since f(2) = 2, we can set up the equation:
8a + 4b + 2 = 2
8a + 4b = 0.
Next, let's divide f(x) by (x - 1) and see what the remainder is.
We know that when f(x) is divided by (x - 1), the remainder is -2. This means that f(1) = -2.
To find f(1), substitute x = 1 into f(x) and solve for it:
f(1) = a(1)^3 + b(1)^2 + 3(1) - 4
= a + b + 3 - 4
= a + b - 1.
Since f(1) = -2, we can set up the equation:
a + b - 1 = -2
a + b = -1.
Now we have a system of two equations:
8a + 4b = 0
a + b = -1.
To solve this system of equations, we can use substitution or elimination method.
Let's use the elimination method:
Multiply the second equation by 4 to make the coefficients of b the same:
4(a + b) = 4(-1)
4a + 4b = -4.
Now, subtract the second equation from the first equation:
8a + 4b - (4a + 4b) = 0 - (-4)
8a + 4b - 4a - 4b = 0 + 4
4a = 4
a = 1.
Substitute the value of a in the second equation:
1 + b = -1
b = -2.
Thus, a = 1 and b = -2.
Therefore, the values of a and b are 1 and -2 respectively.