Consider the polynomial f(x)=17x4+21x3+60x2+Ax+B. Suppose that for every root λ of f(x)=0, 1/λ is also a root of f(x)=0. What is the value of A+B?
If two of the roots are m and n,
f(x) = (x-m)(x-n)(mx-1)(nx-1)
B = mn
17 = mn
so, B=17
mn*1/m + mn*1/n + m*1/m*1/n + n*1/m*1/n = -A/17
n + m + 1/m + 1/n = -A/17
But, we also know that the sum of the roots is -21/17, so
A = 21
A+B = 38
As it happens, all he roots are complex, but they are reciprocals in pairs!
Hmmm. Getting B is ill written.
B/17 = product of all roots
m*n*1/m*1/n = 1
B/17 = 1
B = 17
To find the value of A + B, we need to determine the relationship between the roots of the polynomial f(x) = 17x^4 + 21x^3 + 60x^2 + Ax + B and the roots of f(x) = 0.
Let's assume that λ is a root of f(x) = 0. According to the given condition, 1/λ is also a root of f(x) = 0. Therefore, we can write the polynomial as:
f(x) = (x - λ)(x - 1/λ)(x - α)(x - β), where α and β are the other two roots.
Expanding the polynomial, we get:
f(x) = (x^2 - (λ + 1/λ)x + 1)(x^2 - (α + β)x + αβ)
Comparing this with the original polynomial, we can equate the corresponding coefficients:
17 = αβ
21 = -(λ + 1/λ) - (α + β)
60 = (λ + 1/λ)(α + β) - αβ
To solve these equations, we can substitute αβ = 17 into the third equation:
60 = (λ + 1/λ)(α + β) - 17
Now, let's consider the condition that 1/λ is also a root. So, we have:
f(1/λ) = 0
Substituting the given polynomial equation with x = 1/λ:
17(1/λ)^4 + 21(1/λ)^3 + 60(1/λ)^2 + A(1/λ) + B = 0
Simplifying this expression:
17/λ^4 + 21/λ^3 + 60/λ^2 + A/λ + B = 0
Multiplying throughout by λ^4 to eliminate the denominators:
17 + 21λ + 60λ^2 + Aλ^3 + Bλ^4 = 0
Comparing the coefficients with the original polynomial, we can equate them:
17 = λ^4
21 = Aλ^3
60 = Bλ^4
From the first equation, we have:
λ^4 = 17
Taking the fourth root of both sides:
λ = ±√(17)
Now, substituting λ into the second equation:
21 = A(±√(17))^3
21 = A(±17√(17))
Simplifying:
A = ±21 / (17√(17))
So, there are two possible values for A.
Similarly, substituting λ into the third equation:
60 = B(±√(17))^4
60 = B(17)
Simplifying:
B = 60 / 17
Now, we can calculate the sum of A and B:
A + B = ±21 / (17√(17)) + 60 / 17
This gives us the two possible values for A + B.
In conclusion, the value of A + B depends on the sign in front of A and it can be ±21 / (17√(17)) + 60 / 17.