Problem 72 - Algebra and Trigonometry

Question

Find integers that are upper and lower bounds for the real zeros of the polynomial. $$ P(x)=x^{5}-x^{4}+1 $$

Step-by-Step Solution

Answer

The real zeros of \(P(x)=x^5-x^4+1\) are bounded between -1 and 2.

Step 1: Finding Potential Upper Bound

To find an upper bound for the real zeros, we should test potential bounds using polynomial division or bounds theorem. Let's start with testing if 2 is an upper bound by using synthetic division.

Step 2: Synthetic Division for Upper Bound

Perform synthetic division of \(P(x)\) by \(x-2\):\[\begin{array}{c|cccccc}2 & 1 & -1 & 0 & 0 & 0 & 1 \ & & 2 & 2 & 4 & 8 & 16 \ewline ext{ & } &\hline & 1 & 1 & 2 & 4 & 8 & 17 \\end{array}\]Since all the coefficients in the last row are positive, 2 is an upper bound.

Step 3: Finding Potential Lower Bound

Next, let's find a lower bound. Use synthetic division to test if -1 is a lower bound.

Step 4: Synthetic Division for Lower Bound

Perform synthetic division of \(P(x)\) by \(x + 1\):\[\begin{array}{c|cccccc}-1 & 1 & -1 & 0 & 0 & 0 & 1 \ & & -1 & 2 & -2 & 2 & -2 \ewline ext{ & }&\hline & 1 & -2 & 2 & -2 & 2 & -1 \\end{array}\]Since the signs of coefficients in the last row alternate, -1 is a lower bound.

Key Concepts

Upper and Lower Bounds

Finding the upper and lower bounds of a polynomial is crucial to narrowing down where its real zeros may lie. These bounds help us understand the range of possible values for the real roots of the polynomial. To determine these bounds, we use techniques such as polynomial division, particularly synthetic division. The Bound Theorem states that if a polynomial is divided by a binomial of the form \((x-c)\), and all the coefficients of the quotient have consistent signs (all positive or all negative), then \(c\) is an upper (or lower) bound for the real zeros. This means:

• An upper bound for the real zeros results if all numbers in the bottom row (except the remainder) are positive, after performing synthetic division using a positive \(c\).
• A lower bound results if all numbers have alternating signs, indicating a negative \(c\) is being used as part of the division.

Real Zeros

Real zeros of a polynomial are the values of \(x\) that satisfy the equation \(P(x) = 0\). These zeros are crucial in graphing a polynomial since they are the x-intercepts where the graph crosses the x-axis. Finding real zeros typically involves setting the polynomial equal to zero and solving for \(x\). Real zeros may be found using

• Direct substitution: Plugging in values to test if the polynomial equals zero.
• Graphical methods: Observing the plot of the polynomial to find where it intersects the x-axis.
• Analytical methods: Using techniques like synthetic division or the Newton-Raphson method for more complex polynomials.

Understanding where the real zeros are, with the help of bounds, allows for a better grasp of the behavior and roots of the polynomial function.

Synthetic Division

Synthetic division is a simplified form of polynomial division, particularly useful for dividing by linear expressions of the form \(x - c\). It allows us to quickly test potential roots and bounds of polynomials with ease. This process:

• Is simpler than traditional long division because it only involves the coefficients.
• Quickly reveals the remainder, which helps to determine whether the tested value is a zero or not.
• Provides a useful method for verifying the upper and lower bounds of polynomials.

In the exercise, we used synthetic division to test potential bounds, like 2 and -1, in turn efficiently checking if these could restrict where the real zeros of \(P(x)\) might actually appear.

Polynomial Division

Polynomial division is the foundation for understanding both synthetic division and the behavior of polynomial zeros and bounds. It extends the concept of division from numbers to algebraic expressions:

• Traditional polynomial division follows a process similar to long division with numbers, dividing the polynomial into a divisor repeatedly.
• This method allows for remainders, which can be critical in determining the limitations and behavior of the function.
• It also helps break down complex polynomials into simpler components, potentially identifying factors relating to the zeros of the function.

Understanding polynomial division not only assists in manually computing potential bounds and zeros, but also in comprehending more advanced mathematical concepts related to polynomials.