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

In summary, we used the given equations to find the values of a, b, c, and d by equating like coefficients and creating an augmented matrix. By performing row operations, we obtained the values of a=9, b=-6, c=4, and d=-5.
  • #1
karush
Gold Member
MHB
3,269
5
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
 
Mathematics news on Phys.org
  • #2
From the first equation, we have be equating like coefficients:

\(\displaystyle 10a=90\)

\(\displaystyle 11=3c-1\)

\(\displaystyle -4=b+2\)

\(\displaystyle -17d=85\)

Solve each of these to find the values in question. :)
 
  • #3
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.
 
  • #4
mahalo much

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

but ran into fog banks
 
  • #5
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:

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

Now perform:

\(\displaystyle \frac{1}{10}R_1,\,\frac{1}{3}R_3,\,-\frac{1}{17}R_4\)

to obtain:

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

1. What is the purpose of solving a polynomial scale?

The purpose of solving a polynomial scale is to find the values of the variables a, b, c, and d that satisfy the given polynomial equation. This allows us to graph the polynomial function and understand its behavior and properties.

2. What are the steps to solve a polynomial scale?

The steps to solve a polynomial scale are:

  • 1. Set the polynomial equation equal to zero.
  • 2. Use factoring or the quadratic formula to solve for the roots of the polynomial.
  • 3. Write the polynomial in factored form.
  • 4. Use the values of the roots to find the values of a, b, c, and d.
  • 5. Check your solution by plugging in the values of a, b, c, and d into the original equation.

3. Can all polynomial scales be solved?

No, not all polynomial scales can be solved analytically. Some polynomials have irrational or complex roots which cannot be expressed in terms of real numbers. In these cases, numerical methods or approximation techniques can be used to find approximate solutions.

4. How does solving a polynomial scale relate to finding the roots of a polynomial?

Solving a polynomial scale involves finding the values of the variables that satisfy the polynomial equation, which are also known as the roots of the polynomial. The roots of a polynomial are the x-intercepts of the graph, where the polynomial crosses the x-axis.

5. What are some real-life applications of solving polynomial scales?

Solving polynomial scales has many real-life applications, including:

  • 1. In engineering, polynomial scales are used to model and analyze physical systems such as circuits and structures.
  • 2. In finance, polynomial scales are used to calculate the interest earned on investments and the growth of a company's assets.
  • 3. In statistics, polynomial scales are used to fit curves to data and make predictions.
  • 4. In computer graphics, polynomial scales are used to create smooth and realistic animations.

Similar threads

MHB Roots of Polynomial: Find $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$
    • General Math
Replies
1
Views
741
MHB Solving g(x)-h(x) <0: Find a, b, c, d
    • General Math
Replies
2
Views
884
MHB Adam's question via email about Laplace Transforms
    • General Math
Replies
4
Views
10K
MHB Find the scale factor of triangle ABC to triangle DEF
    • General Math
Replies
5
Views
2K
MHB Find Solutions to a+b+c+d=4, a^2+b^2+c^2+d^2=6, a^3+b^3+c^3+d^3=94/9 in [0,2]
    • General Math
Replies
2
Views
748
Use Remainder theorem to find factors of ##(a-b)^3+(b-c)^3+(c-a)^3##
    • Precalculus Mathematics Homework Help
Replies
4
Views
844
MHB Finding the Largest Root of a Polynomial Using Synthetic Division
    • General Math
Replies
1
Views
840
MHB Evaluate the constant in polynomial function
    • General Math
Replies
7
Views
897
MHB Prove No Integers Solve $ax^3+bx^2+cx+d=1$ for x=19,2 for x=62
    • General Math
Replies
3
Views
851
MHB Find Product: $(a^4+b^4+c^4)$ and $(a^6+b^6+c^6)$
    • General Math
Replies
3
Views
802

Hot Threads

Recent Insights

Back
Top