Problem 76 - Precalculus Mathematics for Calculus

Question

Find integers that are upper and lower bounds for the real zeros of the polynomial. $$P(x)=2 x^{3}-3 x^{2}-8 x+12$$

Step-by-Step Solution

Answer

Upper bound: 3, Lower bound: -4.

Step 1: Understand the Problem

We need to find integers that are upper and lower bounds for the real zeros of the polynomial \(P(x) = 2x^3 - 3x^2 - 8x + 12\). This means identifying numbers beyond which there are no real zeros.

Step 2: Use the Degree of the Polynomial

The degree of the polynomial \(P(x)\) is 3. This means that the polynomial can have at most 3 real roots, including multiplicities.

Step 3: Apply the Upper Bound Test using Synthetic Division

Start with a positive integer (e.g. 1) and use synthetic division to test.1. Divide the coefficients \( [2, -3, -8, 12] \) by \(x-1\): - Bring down the 2 - Multiply 2 by 1, add to -3 to get -1 - Multiply -1 by 1, add to -8 to get -9 - Multiply -9 by 1, add to 12 to get 32. The final result is 3, which is positive. So 1 is not an upper bound.3. Repeat for 2: - Final result is positive, showing 2 is not an upper bound4. Repeat for 3: - Final result is negative, indicating 3 is an upper bound.

Step 4: Apply the Lower Bound Test using Synthetic Division

Start with a negative integer (e.g. -1) using synthetic division.1. Divide by \(x+1\): - Final result is positive, that means -1 is not a lower bound2. Repeat for -2: - Final result is positive, indicating -2 is not a lower bound3. Repeat for -3: - Final result is positive, showing -3 is not a lower bound4. Repeat for -4: - Final result is negative, confirming -4 is a lower bound.

Step 5: Conclusion

After testing, we determine that the integer 3 is an upper bound and -4 is a lower bound for the real zeros of the polynomial \(P(x) = 2x^3 - 3x^2 - 8x + 12\).

Key Concepts

Real Zeros

The real zeros of a polynomial are the values of \( x \) for which the polynomial evaluates to zero. These correspond to the x-intercepts of the polynomial's graph. Understanding real zeros is crucial because they provide key information about the intersection of the polynomial function with the x-axis.

• They help determine the general behavior and shape of the graph.
• Finding real zeros involves solving the equation \( P(x) = 0 \).
• Not all polynomials have real zeros, although they will have as many roots as the polynomial's degree, according to the Fundamental Theorem of Algebra.

Polynomials of degree three, like \( P(x) = 2x^3 - 3x^2 - 8x + 12 \), can have up to three real zeros. These zeros can be distinct or repeated, affecting the shape of the curve. Real zeros are fundamental in polynomial analysis, as they provide insights into potential factors and possible graph intersections.

Synthetic Division

Synthetic division is a streamlined method for dividing polynomials, particularly useful when dividing by a linear factor of the form \( x - c \). It is much simpler than traditional polynomial division and provides quick results, especially when verifying potential roots and bounds.

• Set up the division table using the coefficients of the polynomial.
• Bring down the leading coefficient as the initial entry in the result row.
• Multiply this result by the divisor's root and add it to the next coefficient.
• Continue this process until all coefficients are processed.

Synthetic division not only helps check for real zeros but can also verify if a number is an upper or lower bound for the polynomial’s real zeros.
This is done by analyzing the sequence of resulting coefficients during the division process. For example, when testing \( x - c \), if all results are positive, \( c \) serves as an upper bound. Conversely, if all results alternate in sign, you may have a candidate for a lower bound.

Polynomial Roots Test

The Polynomial Roots Test (often referenced alongside synthetic division) involves finding potential rational roots using the Rational Root Theorem and confirming them through division. It is a systematic way to test possible roots that are consistent with the polynomial's structure.

• First, identify potential rational roots using the possible combinations of the lead coefficient and constant term.
• Then, use synthetic division to verify which candidate makes the polynomial equal zero.
• If synthetic division results in a remainder of zero, the tested value is a root of the polynomial.

The test aids in narrowing down promising candidates for real zeros and facilitates understanding of the polynomial's behavior.
It is often paired with identifying upper and lower bounds, creating a powerful strategy to analyze and solve polynomial equations effectively.