+537 1. There are two values of k for which the cubic polynomial 2x^3 - 9x^2 + 12x - k has a double root. What is the sum of those values?
2. Find the sum of all values for q for which the polynomial x^3 - 12x^2 + qx - 64 has all nonnegative real roots.
3. The fourth-degree polynomial P(x) satisfies P(1)=1, P(2)=2, P(3)=3, P(4)=4, and P(5)=125.What is P(6)?
4. Let a>0, and let P(x) be a polynomial with integer coefficients such that P(1) = P(3) = P(5) = P(7) = a, and P(2) = P(4) = P(6) = P(8) = -a. What is the smallest possible value of a?
+128407 3. The fourth-degree polynomial P(x) satisfies P(1)=1, P(2)=2, P(3)=3, P(4)=4, and P(5)=125.What is P(6)?
A fourth degree polynomial has the form ax^4 + bx^3 + cx^2 + dx + e
We have the following equations
a + b + c + d + e = 1
16a + 8b + 4c + 2d + e = 2
81a + 27b + 9c + 3d + e = 3
256a + 64b + 16c + 4d + e = 4
625a + 125b + 25c + 5d + e = 125
This system is a little lengthy to solve....Wolfram Alpha gives the following solutions
a = 5 b = -50 c = 175 d = -249 e = 120
So.... P(6) = 5(6)^4 - 50(6)^3 + 175(6)^2 - 249(6) + 120 = 606
+128407 1. There are two values of k for which the cubic polynomial 2x^3 - 9x^2 + 12x - k has a double root. What is the sum of those values?
Let the double root be " a " and the other root be " b"
And from Vieta, we have that
a + a + b = 2a + b = 9/2 → b = 9/2 - 2a (1)
a^2 + ab + ab = a^2 + 2ab = 6 (2)
a^2b = k/2 → 2a^2b = k (3)
Sub (1) into (2) and we have that
a^2 + 2a ( 9/2 - 2a) = 6
a^2 + 9a - 4a^2 = 6
-3a^2 + 9a - 6 = 0
3a^2 - 9a + 6 = 0
a^2 - 3a + 2 = 0
(a - 2) (a - 1) = 0
So a = 2 or a = 1
If a = 1, then b = (9/2) - 2a = 9/2 -2(1) = 5/2
And k = 2(1)^2 (5/2) = 5
And if a = 2, then b = (9/2) -2(2) = (9/2) - 4 = 1/2
And k = 2(2)^2 ( 1/2) = 4
So the sum = 4 + 5 = 9
