Problem 85 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

Polynomial is positive at \(x = -2\); thus, no real zeros are less than \(-2\).

Step 1: Substitute the Bound

To prove that there are no real zeros less than -2 for the polynomial \(P(x) = x^4 + x^3 - x^2 + 3\), we start by substituting \(x = -2\) into the polynomial. Calculate \(P(-2)\) to examine the sign of the result.

Step 2: Substitute and Calculate

Substituting \(-2\) into the polynomial gives: \[ P(-2) = (-2)^4 + (-2)^3 - (-2)^2 + 3. \] Simplify and compute: \[ 16 - 8 - 4 + 3 = 7. \] Hence, \(P(-2) = 7\). This output is positive.

Step 2: Evaluate and Determine Sign

Since \(P(-2) > 0\), it suggests no roots at \(x = -2\). To further validate, consider the interval \((-\infty, -2] \), and if \(P(x) > 0\) on this interval, it supports that no zeros are less than \(-2\).

Step 4: Test the Behavior of Polynomial

Check the leading term \(x^4\) in \(P(x)\), which dictates the behavior as \(x\) approaches \(-\infty\). For very negative values of \(x\), the dominant term is \(x^4\), which results in \(P(x)\) remaining positive, supporting the boundedness theorem.

Key Concepts

Polynomial Zeros

A zero of a polynomial is a value of the variable for which the polynomial equals zero. In simpler terms, if you plug in this value into the polynomial equation, it will result in 0. These values are crucial because they tell us where the graph of the polynomial will intersect the x-axis.
To find the zeros of a polynomial, we typically solve the equation by setting it equal to zero and finding solutions for the variable. However, the Boundedness Theorem helps by restricting where real zeros can be.
In our problem, the goal was to show that no real zeros of the polynomial \(P(x) = x^4 + x^3 - x^2 + 3\) exist below \(-2\). By evaluating \(P(-2)\) and finding a positive result, it indicates the function is above the x-axis at this point and does not cross to produce a zero there.

Polynomial Behavior

The behavior of polynomials gives us insights into how they behave over different intervals. When analyzing polynomial behavior, we often look at the endpoints and intervals to understand where it is increasing, decreasing, or where it might have zeros.
For \(P(x) = x^4 + x^3 - x^2 + 3\), evaluating at \(x = -2\) showed us that the polynomial is positive at this point. This step was crucial because it signaled that in intervals around \(-2\), particularly to its left, \(P(x)\) is likely to be positive, which supports the conclusion that there are no zeros less than \(-2\).
Studying this behavior helps to predict the overall shape and interactions of the polynomial graph within specific regions.

Leading Term Analysis

In polynomials, the leading term is the term with the highest power of the variable. It is important because it dictates the end behavior of the polynomial. For \(P(x) = x^4 + x^3 - x^2 + 3\), the leading term is \(x^4\).
This term suggests that as \(x\) tends to infinity in either direction (very large positive or negative), the dominant behavior of the polynomial is driven by \(x^4\). Since \(x^4\) is always non-negative, this leads to the conclusion that for very negative values, \(P(x)\) remains positive.
Thus, checking the leading term confirms the behavior described by the Boundedness Theorem, ensuring that the outcome aligns with no zeros being found for x values below \(-2\). By focusing on the leading term, we simplify understanding complex polynomial behavior.