Find the remainder when the function f(x)=x^3-6x^2-x+30 is divided by (4x-1)
To find the remainder when a polynomial function is divided by another polynomial, we can use polynomial division.
1. First, arrange the terms of the dividend (f(x)) in descending order of powers of x:
f(x) = x^3 - 6x^2 - x + 30
2. Check the degree of the divisor (4x - 1) and the dividend. Since the degree of the divisor is 1 and the degree of the dividend is 3, we need to include the missing degree 2 term in the dividend:
f(x) = x^3 - 6x^2 + 0x - x + 30
3. Divide the first term of the dividend (x^3) by the first term of the divisor (4x) to get the quotient term. The quotient term is (x^3 / 4x) = (1/4)x^2.
4. Multiply the divisor (4x - 1) by the quotient term (1/4)x^2 and subtract it from the dividend:
(x^3 - 6x^2 + 0x - x + 30) - (1/4)x^2 × (4x - 1)
Expanding the multiplication and simplifying, we get:
(x^3 - 6x^2 + 0x - x + 30) - (x^3 - (1/4)x^2)
This simplifies to:
(0x^3 - (1/4)x^2 - x + 30)
5. Repeat steps 3 and 4 with the new simplified dividend.
Divide the first term of the new simplified dividend (0x^3) by the first term of the divisor (4x) to get the quotient term. The quotient term is (0x^3 / 4x) = 0.
Multiply the divisor (4x - 1) by the quotient term 0 and subtract it from the dividend:
(0x^3 - (1/4)x^2 - x + 30) - (0x^3 - 0)
This simplifies to:
(- (1/4)x^2 - x + 30)
6. Repeat steps 3 and 4 with the new simplified dividend.
Divide the first term of the new simplified dividend (-(1/4)x^2) by the first term of the divisor (4x) to get the quotient term. The quotient term is (-(1/4)x^2 / 4x) = -(1/16)x.
Multiply the divisor (4x - 1) by the quotient term (-(1/16)x) and subtract it from the dividend:
(- (1/4)x^2 - x + 30) - (-(1/16)x × (4x - 1))
Expanding the multiplication and simplifying, we get:
(- (1/4)x^2 - x + 30) - (-(1/16)x^2 + (1/16)x)
This simplifies to:
(- (3/16)x^2 - (17/16)x + 30)
We have now reached a simplified dividend that has a degree less than the divisor. So, this is our remainder:
Remainder = -(3/16)x^2 - (17/16)x + 30
Therefore, when the function f(x) = x^3 - 6x^2 - x + 30 is divided by (4x - 1), the remainder is -(3/16)x^2 - (17/16)x + 30.