Three roots of f(x)=x^4−2x^3+ax^2+bx+c are −5, −3 and 4. What is the value of a+b+c?
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To find the value of a+b+c, we need to first find the coefficients of the quadratic term (ax^2), the linear term (bx), and the constant term (c) in the given polynomial.
We are given that the polynomial f(x) has three roots: -5, -3, and 4. Since these are the roots of the polynomial, we can write three equations using these roots.
Let's write the equations using the given roots:
For x = -5:
f(-5) = (-5)^4 - 2(-5)^3 + a(-5)^2 + b(-5) + c = 0
For x = -3:
f(-3) = (-3)^4 - 2(-3)^3 + a(-3)^2 + b(-3) + c = 0
For x = 4:
f(4) = (4)^4 - 2(4)^3 + a(4)^2 + b(4) + c = 0
Now, we have a system of three equations with three unknowns (a, b, and c). We can solve this system to find the values of a, b, and c.
Using the given values of the roots, we can simplify the equations:
(-5)^4 - 2(-5)^3 + a(-5)^2 + b(-5) + c = 0
625 + 250a - 125b + 5c = 0 -- Equation 1
(-3)^4 - 2(-3)^3 + a(-3)^2 + b(-3) + c = 0
81 - 54a + 9b + c = 0 -- Equation 2
(4)^4 - 2(4)^3 + a(4)^2 + b(4) + c = 0
256 - 128a + 16b + 4c = 0 -- Equation 3
We now have a system of three equations:
625 + 250a - 125b + 5c = 0 -- Equation 1
81 - 54a + 9b + c = 0 -- Equation 2
256 - 128a + 16b + 4c = 0 -- Equation 3
By solving this system of equations, we can determine the values of a, b, and c. Once we have these values, we can simply add them together to find the value of a+b+c.