We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
112
2
avatar+363 

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+104723 
+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+104723 
+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

30 Online Users

avatar
avatar
avatar