Intermediate Value Theorem and Synthetic Division

In summary, the given problem asks if the Intermediate Value Theorem guarantees a real solution between two given points for a given equation. Two methods are discussed to determine the sign of the given points in the equation: evaluating the function at the points and using synthetic division. The first method shows that the points do not satisfy the equation, while the second method uses the Remainder Theorem to find the value of the function at the points. However, it is important to note that the quotient in synthetic division is only meaningful when the divisor is not 0, making the information about the sign of the original polynomial unreliable. Instead, the remainder can be used to find the sign of the function at the given points.
  • #1

opus

Gold Member
717
131
Say I have a given problem that states:

Does the Intermediate Value Theorem guarantee that the following equation has a real solution between ##(\frac{7}{2})## and ##(\frac{9}{2})##?
$$3x^4-27x^3+177x^2+1347x+420=0$$

Now what I want to do is determine the sign of x=##(\frac{7}{2})## and x=##(\frac{9}{2})## at the given equation.

I can do this two ways: one way is to just evaluate ##f(7/2)## and ##f(9/2)##. This gives two positive results, so the Intermediate Value Theorem cannot guarantee that there's a zero between these points. This leads me into my fist question:
1) In plugging the two given x values into the given equation, the left side does not equal zero. So this is telling me that the two given x values are not solutions to the equation (meaning the graph doesn't cross the x-axis at these points). However there is a point where the graph does have the x-value of x=##(\frac{7}{2})## and x=##(\frac{9}{2})## and at these points, the graph is positive. Is this a true statement?

Another way to do this is to use synthetic division, dividing the polynomial by k=##(7/2)## and k=##(9/2)## This works because of the Remainder Theorem- where if we divide a polynomial ##p(x)## by ##(x-k)##, the remainder is ##p(k)##. Since x=k, we find the value of the function ##3x^4-27x^3+177x^2+1347x+420=0## evaluated at x=##(\frac{7}{2})## and x=##(\frac{9}{2})##. In this case, the remainders are both positive, and is the second part of my question.

When dividing polynomials, if the remainder is positive, does that mean that the entire quotient is positive? In other words, if the final digit in the synthetic division is positive, is the entire quotient positive? This question doesn't seem very intuitive to me as it's not with concrete number values, but with expressions with variables and graphs that can fluctuate between positive and negative values.
 
Mathematics news on Phys.org
  • #2
1) Yes.

2) No. The quotient is a polynomial, plus a fraction whose denominator is the factor ##(x-k)##. In particular, if your synthetic division is:
$$\begin{array}{r|rrrrr}\frac{7}{2} & 3 & -27 & 177 & 1347 & 420\\ & & \frac{21}{2} & -\frac{231}{4} & \frac{3339}{8} & \frac{98805}{16}\\\hline & 3 & -\frac{33}{2} & \frac{477}{4} & \frac{14115}{8} & \frac{105525}{16}\end{array}$$
then the quotient only means that, when ##\left(x - \frac{7}{2}\right)## is not 0,
$$3x^4 - 27x^3 + 177x^2 + 1347x + 420 = \left(x - \frac{7}{2}\right)\left(3x^3 - \frac{33}{2}x^2 + \frac{477}{4}x + \frac{14115}{8} + \frac{105525}{16\left(x - \frac{7}{2}\right)}\right)$$
The quotient is meaningless when ##x = k##, since in that particular case, you would be attempting to divide by ##0##. So no information about the sign of the original polynomial can be retrieved this way, without doing a lot more work than just finding the value of the original polynomial when ##x = k## manually.
 
  • Like
Likes opus
  • #3
Thank you for that response. I also didn't know you can write synthetic division out here, so I'll keep that in mind for asking better questions next time.

So for the second part, to try to summarize, can we say that we use synthetic division because in dividing a polynomial by ##(x-k)##, the remainder is equal to ##f(k)## which will give the same solution if were were to just plug in the x=k into the function and evaluate it that way? So in both instances, we are getting the same solution, so the sign of the remainder in synthetic division, is the same sign as the x=k evaluated with the function?

I'm trying to word this in a way that it makes clear how I'm thinking about it. But ideally I want to point at things and speak about them, which is not possible through typing obviously. So I hope what I've said makes sense.
 
  • #4
You've got it. You can use the remainder to find the sign of f(k), since the remainder is f(k), but not necessarily the sign of the quotient, which depends on the value of x.
 
  • Like
Likes opus
  • #5
Thanks!
 
  • Like
Likes slider142

What is the Intermediate Value Theorem?

The Intermediate Value Theorem is a mathematical concept that states if a continuous function has two points with different signs, then there must be at least one point between them where the function equals zero.

What is the significance of the Intermediate Value Theorem?

The Intermediate Value Theorem is important because it can prove the existence of roots or solutions to equations that cannot be solved algebraically. It also helps to ensure the accuracy of numerical methods for solving equations.

What is synthetic division?

Synthetic division is a method for dividing polynomials in which the coefficients of the polynomials are written in a shorthand form and only the numerical values are used in the division process. This method is commonly used to find roots or factors of polynomial equations.

How is synthetic division related to the Intermediate Value Theorem?

Synthetic division is related to the Intermediate Value Theorem because it can be used to test whether a given value is a root or solution of a polynomial equation. By plugging in the given value and using synthetic division, we can determine if the function changes sign and therefore if the Intermediate Value Theorem applies.

What are some common applications of the Intermediate Value Theorem and synthetic division?

The Intermediate Value Theorem and synthetic division are commonly used in fields such as engineering, economics, and physics to model and solve real-world problems. They are also used in calculus to prove the existence of certain mathematical concepts and in computer science for data analysis and optimization.

Suggested for: Intermediate Value Theorem and Synthetic Division

Replies
2
Views
737
Replies
50
Views
3K
2
Replies
47
Views
2K
Replies
40
Views
2K
Replies
1
Views
552
Replies
10
Views
1K
Replies
1
Views
760
Replies
5
Views
586
Replies
1
Views
569
Back
Top