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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?

 Sep 15, 2019
 #1
avatar+106535 
+1

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 

 

 

cool cool cool

 Sep 15, 2019
 #2
avatar+106535 
+1

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  

 

 

 

cool cool cool

 Sep 15, 2019

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