Solve Polynomial Scale: Find a,b,c,d

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Discussion Overview

The discussion revolves around solving for the coefficients a, b, c, and d in a polynomial equation involving the variable s. Participants explore methods to derive these coefficients based on given equations and values of s, with a focus on algebraic manipulation and matrix methods.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant seeks to determine the values of a, b, c, and d such that the polynomial equation holds for all values of s, providing specific equations derived from substituting values of s.
  • Another participant suggests equating coefficients from the polynomial equations to find the values of a, b, c, and d, indicating that solving these equations will yield the necessary coefficients.
  • A different participant points out that four unknowns require four equations, proposing to substitute an additional value for s to generate the needed equation.
  • One participant shares their previous attempts at solving the problem through factoring and considers using a matrix approach, expressing some confusion about the process.
  • A later reply provides an augmented matrix representation of the equations and outlines steps to simplify it, suggesting a method to solve for the coefficients.

Areas of Agreement / Disagreement

Participants generally agree on the need for additional equations to solve for all coefficients, but there is no consensus on the best method to achieve this or on the specific values of a and b.

Contextual Notes

The discussion highlights the challenge of solving a system of equations with multiple unknowns and the reliance on specific values of s to generate sufficient equations. There are unresolved aspects regarding the completeness of the equations provided and the methods suggested.

karush
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what is a b c and d so that all values of s are true

\begin{align}\displaystyle
&f_{15}=\\
&-17d+11s^2-4s+10as^3=(b+2)s+90s^3+(3c-1)s^2+85\\
&-17d+11s^2-6s+10as^3=bs+90s^3+3cs^2-s^2+85\\
&(s=0)\\
&-17d=85 \therefore d=-5\\
&11s^2-6s+10as^3=bs+90s^3+3cs^2-s^2\\
&(s=-1)\\
&11+6-10a=-b-90+3c-1\\
&-10a+b-3c=-18 \\
&(s=1) \\
&11-6+10a=b+90+3c-1\\
&10a-b-3c=-6 \\
&--3c=-24\therefore c=4
\end{align}

what i couldn't get is a and b
 
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From the first equation, we have be equating like coefficients:

$$10a=90$$

$$11=3c-1$$

$$-4=b+2$$

$$-17d=85$$

Solve each of these to find the values in question. :)
 
Or: you have four unknowns, a, b, c, and d so you need four equations, not just three. Take one more values for s, say s= 2, to get one more equation.
 
mahalo much

I tried earlier to do this by some factoring
and a thot a matrix could be used

but ran into fog banks
 
karush said:
mahalo much

I tried earlier to do this by some factoring
and a thot a matrix could be used

but ran into fog banks

You could use an augmented matrix:

$$\left[\begin{array}{cccc|c}10 & 0 & 0 & 0 & 90 \\ 0 & 1 & 0 & 0 & -6 \\ 0 & 0 & 3 & 0 & 12 \\ 0 & 0 & 0 & -17 & 85 \\ \end{array}\right]$$

Now perform:

$$\frac{1}{10}R_1,\,\frac{1}{3}R_3,\,-\frac{1}{17}R_4$$

to obtain:

$$\left[\begin{array}{cccc|c}1 & 0 & 0 & 0 & 9 \\ 0 & 1 & 0 & 0 & -6 \\ 0 & 0 & 1 & 0 & 4 \\ 0 & 0 & 0 & 1 & -5 \\ \end{array}\right]$$
 

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