Solve 4th Degree Polynomial w/ Slope Constraints

In summary: The first row is all zeros, so it doesn't contribute to the solution. The second and third rows are the equations you found earlier. The fourth and fifth rows are the equations from taking the derivatives and plugging in the x-values given for the slope. Solving this augmented matrix will give you the values of a, b, c, d, and e for the polynomial p(x) = ax^4 + bx^3 + cx^2 + dx + e.
  • #1
crombiecutie105
3
0

Homework Statement



the question states: Find the coefficients of the fourth degree polynomial:
p(x) = ax^4 + bx^3 + cx^2 + dx + e whose graph goes through the points (0,0), (1,1), (-1,3) and whose slope at x=-1 is 20 and x=1 is 9.

Homework Equations





The Attempt at a Solution



i started by putting it into an augmented matrix, and solving...
my matrix was:

0 0 0 0 1 | 0
1 1 1 1 1 | 1
1 -1 1 -1 1 | 3

when i solved for the coefficients, i got a = 0, b = -1, c = 2, d = 0, and e = 0.
i'm not sure how to incorporate the information given by the slopes.

i know that you can take the derivative to find slope.. and i took the derivative of the polynomial and got:

4ax^3 + 3bx^2 + 2cx + d

but I'm not really sure where to go from there. do i plug in -1 for x and set it equal to 20 and 1 for x and set it equal to 9? if i do that i get the equations:

-4a + 3b - 2c + d = 20 and
4a + 3b + 2c + d = 9

by now i think I'm starting to make things up though! :wink:
can anyone offer any hints? thank you!
 
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  • #2
crombiecutie105 said:
but I'm not really sure where to go from there. do i plug in -1 for x and set it equal to 20 and 1 for x and set it equal to 9? if i do that i get the equations:

Yes, you do. I didn't check your calculations, but the next step is obviously to solve a system of equations.
 
  • #3
i set up a system of equations for those two equations and with only 2 equations and 4 variables i was unsure of how to proceed. i ended up with the matrix:

-4 3 -2 1 | 20
0 6 0 2 | 29

i'm just confused as to how the two steps are connected... like the first time i solved for the coefficients and this time. can i combine my results somehow?
 
  • #4
I don't follow. Did you set up a system of 5 equations with 5 unknowns? Did you try to solve it?
 
  • #5
ok, i think i got it now. i didnt think i could set up all 5 equations before because the first set of equations had 5 variables and the second set of equations had 4. i put a zero as a place holder though, and solved this matrix:

0 0 0 0 1 | 0
1 1 1 1 1 | 1
1 -1 1 -1 1 | 3
-4 3 -2 1 0 |20
4 3 2 1 0 |9

i think I'm good from here. thanks for your help!
 
  • #6
The matrix looks okay.
 

FAQ: Solve 4th Degree Polynomial w/ Slope Constraints

What is a 4th degree polynomial?

A 4th degree polynomial is a mathematical expression that contains terms with variables raised to the 4th power. It can be written in the form of ax^4 + bx^3 + cx^2 + dx + e, where a, b, c, d, and e are coefficients and x is the variable.

What does it mean to solve a 4th degree polynomial with slope constraints?

Solving a 4th degree polynomial with slope constraints means finding the values of the coefficients that satisfy both the polynomial equation and a set of constraints on the slope of the polynomial at certain points. This can involve finding the maximum or minimum values of the polynomial or ensuring the polynomial has a certain shape at specific points.

How do you solve a 4th degree polynomial with slope constraints?

To solve a 4th degree polynomial with slope constraints, you can use a variety of methods such as substitution, graphing, or using calculus techniques. The specific method will depend on the constraints given and the type of polynomial being solved.

What are some real-world applications of solving 4th degree polynomials with slope constraints?

Solving 4th degree polynomials with slope constraints can be useful in many fields such as engineering, physics, and economics. For example, in engineering, it can be used to optimize the shape of a structure to withstand certain forces, while in economics, it can be used to determine the maximum profit for a business based on various production constraints.

Are there any special techniques for solving 4th degree polynomials with slope constraints?

Yes, there are some special techniques for solving 4th degree polynomials with slope constraints such as the method of Lagrange multipliers and the method of undetermined coefficients. These methods can be used to efficiently solve the polynomial while satisfying the given constraints.

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