Problem 87 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

Real zeros of \( P(x) \) are not greater than 1.

Step 1: Understand the Problem and Boundedness Theorem

We are given the polynomial \( P(x) = 3x^4 + 2x^3 - 4x^2 + x - 1 \) and need to demonstrate that it has no real zeros greater than 1 using the boundedness theorem, which implies that if a polynomial evaluated at a point gives a positive result or if the synthetic division gives no sign changes, there aren't zeros beyond that point.

Step 2: Synthetic Division with Upper Bound

Perform synthetic division of \( P(x) \) by \( x - 1 \) to test if there are any positive signs across all coefficients in the division process. If the signs are consistent and non-negative, 1 is an upper bound, indicating no zeros exist beyond this point.

Step 3: Executing Synthetic Division

Using synthetic division, divide:\[\begin{array}{r|rrrrr} 1 & 3 & 2 & -4 & 1 & -1 \ & & 3 & 5 & 1 & 2 \\hline & 3 & 5 & 1 & 2 & 1 \\end{array}\]Since all coefficients 3, 5, 1, 2, and 1 are positive (non-negative), there are no sign changes.

Step 4: Conclude with Boundedness Theorem

The result from synthetic division shows all non-negative coefficients, confirming no sign changes. This means by the boundedness theorem, all real zeros are less than or equal to 1 and there is no real zero greater than 1.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve a sum of powers of a variable with constant coefficients. They are crucial in algebra and calculus due to their structured nature, making them easier to analyze than more complex expressions. A typical polynomial function is of the form:\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x+ a_0 \]where \( a_n, a_{n-1}, \ldots, a_0 \) are constants, and \( n \) is a non-negative integer representing the degree of the polynomial.

• The degree of the polynomial is determined by the highest power of \( x \), such as 4 in the polynomial function \( 3x^4 + 2x^3 - 4x^2 + x - 1 \).
• Polynomial functions are continuous and smooth, meaning they have no breaks or sharp corners in their graphs.
• They have applications in various fields like physics, engineering, economics, and biology.

This understanding serves as a foundation for analyzing real zeros and performing operations like synthetic division.

Synthetic Division

Synthetic division is a simplified method of polynomial division, particularly useful for dividing by binomials of the form \( x - c \). It provides a quick way to find the remainder and the quotient without writing out the full polynomial long division.To perform synthetic division:

• Identify \( c \) from the divisor \( x - c \); here, \( c = 1 \).
• Write down the coefficients of the polynomial \([3, 2, -4, 1, -1]\).
• Bring down the first coefficient (3 in this case).
• Multiply \( c \) by the number obtained below the line and add this to the next coefficient.

It's essential in confirming bounds of real zeros by checking the signs of the coefficients obtained after division. The sequence \([3, 5, 1, 2, 1]\) present in the step confirms no sign change, integral to the upper bound test for roots.

Real Zeros

Real zeros of a polynomial are the values of \( x \) where the polynomial evaluates to zero. These zeros correspond to the x-intercepts of the polynomial on a graph and are essential in understanding the behavior and properties of the polynomial.To find real zeros:

• Using methods like factoring (when possible),
• Employing the Rational Root Theorem, and
• Using numerical approximation techniques when exact methods fail.

The boundedness theorem connects with real zeros by confirming if certain values could possibly be real zeros. Understanding real zeros helps in predicting how the function will behave across the number line by identifying meaningful transitions in the graph.

Upper Bound Test

The upper bound test is a part of the boundedness theorem, which is used to confirm the possible extent of real zeros of a polynomial. When performing synthetic division on a polynomial with \( x - c \), and if all coefficients in the resulting sequence are non-negative, \( c \) is considered an upper bound.The test works effectively because:

• If all resulting coefficients are positive, then no larger real zero exists beyond the tested value.
• This test is crucial in narrowing down the range of possible real zeros.
• It can also simplify the process of finding actual zeros by limiting unnecessary calculations.

In the given problem, \( c = 1 \) proves to be the upper bound as all coefficients \([3, 5, 1, 2, 1]\) in the synthetic division step were positive, confirming that no real zeros are greater than 1.