When x^4 + ax^3 + bx +c is divided by (x–1), (x+1), and (x+2), the remainders are 14, 0, and –16 respectively. Find the values of a, b, and c.
do a little synthetic division. The remainder when divided by (x-1) is a+b+c+1. So, one equation is
a+b+c+1 = 14
do the others, then solve the three equations for a,b,c.
first division -> a+b+c-1 = 14
second division -> c-a-b+1 = 0
third division-> have to go, continue ;)
To find the values of a, b, and c, we can use the Remainder Theorem. According to the Remainder Theorem, if a polynomial f(x) is divided by x - r, where r is a root of f(x), then the remainder will be zero.
We are given that when the polynomial x^4 + ax^3 + bx + c is divided by (x - 1), (x + 1), and (x + 2), the remainders are 14, 0, and -16 respectively. This means that the polynomial is divided by these factors, resulting in zero remainders.
1. Dividing x^4 + ax^3 + bx + c by (x - 1) gives a remainder of 14:
When we divide x^4 + ax^3 + bx + c by x - 1, the resulting quotient will be a cubic polynomial plus a constant term, and the remainder will be a constant term. So we have the equation:
x^4 + ax^3 + bx + c = (x - 1) * (some cubic polynomial + constant) + 14
Expanding the right side of the equation, we get:
x^4 + ax^3 + bx + c = x * (some cubic polynomial + constant) - 1 * (some cubic polynomial + constant) + 14
x^4 + ax^3 + bx + c = x * (some cubic polynomial) - (some cubic polynomial) + x * (constant) - (constant) + 14
Comparing coefficients on both sides of the equation, we get:
a = 1 (coefficient of x^3)
b - (some cubic polynomial) = 0 (coefficient of x)
c - (constant) + 14 = 0 (constant term)
Simplifying the equations, we find:
a = 1
b = (some cubic polynomial)
c = -14 + (constant)
2. Dividing x^4 + ax^3 + bx + c by (x + 1) gives a remainder of 0:
Following the same steps as above, we have the equation:
x^4 + ax^3 + bx + c = (x + 1) * (some cubic polynomial + constant)
Expanding the right side of the equation, we get:
x^4 + ax^3 + bx + c = x * (some cubic polynomial + constant) + 1 * (some cubic polynomial + constant)
Comparing coefficients on both sides of the equation, we find:
a + (some cubic polynomial) = 0 (coefficient of x^3)
b + (some cubic polynomial) = 0 (coefficient of x)
c + (constant) = 0 (constant term)
Simplifying the equations, we find:
a + (some cubic polynomial) = 0
b + (some cubic polynomial) = 0
c + (constant) = 0
3. Dividing x^4 + ax^3 + bx + c by (x + 2) gives a remainder of -16:
Following the same steps as above, we have the equation:
x^4 + ax^3 + bx + c = (x + 2) * (some cubic polynomial + constant) - 16
Expanding the right side of the equation, we get:
x^4 + ax^3 + bx + c = x * (some cubic polynomial + constant) + 2 * (some cubic polynomial + constant) - 16
x^4 + ax^3 + bx + c = x * (some cubic polynomial) + 2 * (some cubic polynomial) + x * (constant) + 2 * (constant) - 16
Comparing coefficients on both sides of the equation, we find:
a + (some cubic polynomial) = 0 (coefficient of x^3)
b + 2 * (some cubic polynomial) = 0 (coefficient of x)
c + 2 * (constant) - 16 = 0 (constant term)
Simplifying the equations, we find:
a + (some cubic polynomial) = 0
b + 2 * (some cubic polynomial) = 0
c + 2 * (constant) = 16
By solving these three sets of equations simultaneously, we can determine the values of a, b, and c.