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. (4 marks)
Please help, I understand what to do when there's only 2 variables but not with 3. Thank you!
To find the values of a and b, we can use the concept of polynomial remainders and the Remainder Theorem.
Let's start by applying the Remainder Theorem to the first condition. According to the Remainder Theorem, if f(x) is divided by (x - c), then the remainder will be equal to f(c), where c is a constant.
In this case, when f(x) is divided by (x - 2), the remainder is given as 2. So, we can write f(2) = 2.
Substituting x = 2 into the expression for f(x):
f(2) = a(2^3) + b(2^2) + 3(2) - 4
2 = 8a + 4b + 6 - 4
Simplifying the equation:
8a + 4b + 2 = 0
Dividing both sides by 2:
4a + 2b + 1 = 0 -- Equation 1
Similarly, for the second condition, when f(x) is divided by (x - 1), the remainder is -2. So, we can write f(1) = -2.
Substituting x = 1 into the expression for f(x):
f(1) = a(1^3) + b(1^2) + 3(1) - 4
-2 = a + b + 3 - 4
Simplifying the equation:
a + b - 3 = 0 -- Equation 2
Now we have two equations, Equation 1 and Equation 2, with two variables a and b. We can solve these equations simultaneously to find their values.
1. Multiply Equation 2 by 2:
2a + 2b - 6 = 0 -- Equation 3
2. Subtract Equation 3 from Equation 1:
(4a + 2b + 1) - (2a + 2b - 6) = 0
4a + 2b + 1 - 2a - 2b + 6 = 0
2a + 7 = 0
Simplifying and solving for a:
2a = -7
a = -7/2
3. Substituting the value of a in Equation 2:
(-7/2) + b - 3 = 0
b = 3 + (7/2)
b = 13/2
Finally, we have found the values of a and b. a = -7/2 and b = 13/2.