Problem 84 - A Graphical Approach to Precalculus with Limits

Question

Use the boundedness theorem to show that the real zeros of \(P(x)\) satisfy the given conditions. \(P(x)=2 x^{5}-x^{4}+2 x^{3}-2 x^{2}+4 x-4\) no real zero greater than 1

Step-by-Step Solution

Answer

The real zeros of \(P(x)\) are less than or equal to 1.

Step 1: Introduction to the Boundedness Theorem

The Boundedness Theorem suggests that if a polynomial \(P(x)\) has real roots, they must exist within certain boundaries relative to its coefficients. By evaluating the polynomial using synthetic division and observing changes in sign, we can establish such boundaries.

Step 2: Set Up Synthetic Division for \(P(x)\)

To apply the Boundedness Theorem for upper bounds, perform synthetic division of the polynomial \(P(x) = 2x^5 - x^4 + 2x^3 - 2x^2 + 4x - 4\) by \(x-c\). Start with \(c=1\) to check if 1 is an upper bound.

Step 3: Perform Synthetic Division by 1

Write down the coefficients: 2, -1, 2, -2, 4, -4. Using 1 as the divisor, compute the remainder through synthetic division: 1. Bring down the first coefficient (2). 2. Multiply 2 by 1 and add it to -1, giving 1. 3. Multiply 1 by 1 and add it to 2, giving 3. 4. Multiply 3 by 1 and add it to -2, giving 1. 5. Multiply 1 by 1 and add it to 4, giving 5. 6. Multiply 5 by 1 and add it to -4, giving 1.

Step 4: Analyze the Sign of the Remainder

After performing synthetic division, the row shows the sequence: 2, 1, 3, 1, 5, 1. Since all the results are non-negative, this indicates no sign changes in the sequence. Therefore, \(x=1\) is an upper bound for the real roots of \(P(x)\).

Step 5: Conclusion

Since there are no sign changes in the synthetic division row, no real zero of \(P(x)\) can be greater than 1. The Boundedness Theorem confirms \(x=1\) as an upper bound for real zeros.

Key Concepts

Synthetic Division

Synthetic division is a simplified form of polynomial division, especially useful when dividing by a linear expression of the form \(x-c\). It reduces the complexity involved by focusing only on the coefficients of the polynomial.

• Steps to Perform Synthetic Division:
  • Write Down the Coefficients: List out the coefficients of the polynomial in decreasing order of power.
  • Bring Down the First Coefficient: This becomes the starting point of our solution.
  • Multiply and Add: Multiply the divisor with the number in the result line, add it to the next coefficient, and continue this process down the line.

This method is not only faster but also efficient for finding zeros and verifying possible roots defined by the polynomial's factors.

Polynomials

A polynomial is a mathematical expression consisting of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication. Polynomials are fundamental in algebra and come with several properties that make them intriguing to study.

• Standard Form: Polynomials are typically written in descending order of power, like \(ax^n + bx^{n-1} + \ldots + z\).
• Degree: The highest exponent in the polynomial signifies its degree, determining the number of roots it can have (real or complex).
• Roots: A root is a solution where the polynomial equals zero when substituted with a value (i.e., solving \(P(x)=0\)).

Understanding polynomials involves identifying these features and their implications on the behavior of the function.

Upper Bound of Zeros

The upper bound of zeros refers to the highest possible value for which a polynomial equation can have a real root. This upper bound can be identified using synthetic division and the Boundedness Theorem.

• Using Synthetic Division: By performing synthetic division and looking for a sequence of non-negative results, we establish that an upper boundary exists for the polynomial's zeros.
• Boundedness Theorem: This theorem helps predict where zeros can be found, ensuring that none are unexpectedly beyond certain limits based on the polynomial's coefficients.

Finding this upper bound makes solving polynomials more manageable by limiting the region we need to examine for real solutions. It assures us that no root can be greater than this defined boundary, as seen in our example with \(P(x)\).