Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Use synthetic substitution to determine whether the given number is a zero of the polynomial. $$\sqrt{6} ; \quad P(x)=-2 x^{6}+5 x^{4}-3 x^{2}+270$$

Step-by-Step Solution

Answer

Yes, \(\sqrt{6}\) is a zero of the polynomial.

Step 1: Set up Synthetic Substitution

For synthetic substitution, we will substitute \(x = \sqrt{6}\) into the polynomial \(-2x^6 + 5x^4 - 3x^2 + 270\). This involves evaluating the polynomial at \(x = \sqrt{6}\) to determine if it results in zero.

Step 2: Evaluate Each Term of the Polynomial

Calculate each term of the polynomial separately for \(x = \sqrt{6}\):- Calculate \((-2)(\sqrt{6})^6 = -2(\sqrt{6})^{6} = -2(216) = -432\)- Calculate \(5(\sqrt{6})^4 = 5 \times 36 = 180\)- Calculate \(-3(\sqrt{6})^2 = -3 \times 6 = -18\)- The constant term is 270.

Step 3: Add the Evaluated Terms

Sum the results from Step 2:- Adding the results: \(-432 + 180 - 18 + 270\)- Calculating the sum: \((-432) + 180 = -252\), then \(-252 - 18 = -270\), and finally \(-270 + 270 = 0\).

Step 4: Conclusion

Since the sum of the evaluated terms equals zero, \(P(\sqrt{6}) = 0\). This confirms that \(\sqrt{6}\) is a zero of the polynomial.

Key Concepts

Synthetic Substitution

Synthetic substitution is a technique used to evaluate polynomials quickly and efficiently, especially when checking if a value is a zero of the polynomial. Instead of plugging the value into the polynomial equation directly, synthetic substitution allows us to perform a structured algebraic process. This process involves using a table-like format to carry out the multiplications and additions in a concise manner.

To use synthetic substitution:

• Identify the value you are testing, which, in this case, is \(x = \sqrt{6}\).
• Rewrite the polynomial in a descending order of the powers of \(x\).
• For each term of the polynomial, substitute the given \(x\) value and evaluate individually.

By systematically evaluating each term separately, we reduce potential arithmetic errors and streamline the problem-solving process.

Zeros of Polynomials

The zeros (or roots) of a polynomial are the values of \(x\) for which the polynomial equals zero. In other words, these are the solutions to the equation \(P(x) = 0\). Finding these zeros is crucial because they tell us where the polynomial graph intersects the x-axis.

If a value, say \(x_0\), makes a polynomial \(P(x)\) equal to zero—\(P(x_0) = 0\)—then \(x_0\) is a zero of the polynomial. Zeros can be real or complex numbers, and a polynomial of degree \(n\) has exactly \(n\) zeros in the complex plane, counting multiplicities.

In our example, performing synthetic substitution demonstrated that \(\sqrt{6}\) is a zero of the polynomial \(-2x^6 + 5x^4 - 3x^2 + 270\). Being able to determine zeros effectively can help in graphing polynomials and solving polynomial equations.

Evaluating Polynomials

Evaluating a polynomial at a specific value involves substituting the value for the variable in the polynomial and performing the arithmetic calculations. This process helps determine the output of the polynomial for that value or verify if the value is a zero.

To evaluate a polynomial quickly:

• For each term, compute the power of \(x\) and multiply by the coefficient.
• Include all computed terms, including any constant term at the end of the polynomial.
• Combine these terms using addition or subtraction as indicated by the polynomial.

In the provided exercise, evaluating the polynomial \(P(x) = -2x^6 + 5x^4 - 3x^2 + 270\) at \(x = \sqrt{6}\) verified that the entire polynomial simplifies to zero, thus confirming that \(\sqrt{6}\) is a valid zero. Evaluating polynomials is a powerful skill that helps establish the behavior of polynomial functions across different scenarios.