Problem 6 - Precalculus with Calculus Previews

Question

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=8 x^{4}-2 x^{3}+15 x^{2}-4 x-2 $$

Step-by-Step Solution

Answer

The rational zero of the polynomial is \( x = -\frac{1}{2} \).

Step 1: Identify Potential Rational Zeros

To find potential rational zeros, use the Rational Root Theorem. According to the theorem, any rational zero, \( \frac{p}{q} \), must satisfy that \( p \) is a factor of the constant term \(-2\) and \( q \) is a factor of the leading coefficient \(8\). Thus, the possible values for \( p \) are \( \pm 1, \pm 2 \) and for \( q \) are \( \pm 1, \pm 2, \pm 4, \pm 8 \). The potential rational zeros are \( \pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8} \).

Step 2: Test Each Potential Rational Zero

Substitute each of the values from the list into the polynomial function \( f(x) \) and check which ones make \( f(x) = 0 \). Start with the simplest potential zeros.

Step 3: Evaluate Zero Candidates

Calculate \( f(1) = 8(1)^4 - 2(1)^3 + 15(1)^2 - 4(1) - 2 = 8 - 2 + 15 - 4 - 2 = 15 \). Therefore, \( x = 1 \) is not a zero. Continue testing with further candidates.

Step 4: Find a Valid Zero

Continue the process until a valid zero is found. Calculate \( f(-1) = 8(-1)^4 - 2(-1)^3 + 15(-1)^2 - 4(-1) - 2 = 8 + 2 + 15 + 4 - 2 = 27 \). \(-1\) is not a zero either. Proceed similarly for other rational candidates.

Step 5: Confirm Valid Rational Zero

On testing \( x = -\frac{1}{2} \), calculate \[ f\left(-\frac{1}{2}\right) = 8 \left(-\frac{1}{2}\right)^4 - 2 \left(-\frac{1}{2}\right)^3 + 15 \left(-\frac{1}{2}\right)^2 - 4 \left(-\frac{1}{2}\right) - 2 = 0 \]. Thus, \( x = -\frac{1}{2} \) is a zero.

Step 6: Repeat Process for Additional Zeros

After finding a valid zero, apply polynomial long division or synthetic division to divide the polynomial by \((x + \frac{1}{2})\). This simplifies the polynomial, allowing further exploration of additional zeros using the quotient polynomial.

Key Concepts

Polynomial Function

A polynomial function is an expression made up of variables and coefficients, constructed using operations such as addition, subtraction, multiplication, and non-negative integer exponents. In more familiar terms, it is a sum of several terms, each consisting of a coefficient multiplied by a variable raised to an integer power.
For example, in the polynomial function given in the exercise:

• 8 is the coefficient of the highest power term, which is 8x⁴.
• The function also includes other terms: -2x³, 15x², -4x, and -2.
• The constant term, -2, is key when using certain methods like the Rational Root Theorem.

Understanding polynomial functions is crucial as they are used across various areas in mathematics, including calculus, where they represent curves, and algebra, where they help in finding roots or solutions.

Synthetic Division

Synthetic division is a simplified way to divide a polynomial by a linear binomial, such as \(x - c\). This method is preferred over long division for its efficiency and reduced chance for error.
Here’s a basic outline of the process:

• Write down the coefficients of the polynomial.
• Identify the zero of the divisor \(x - c\), which is \(c\). Use this value in the synthetic division.
• Bring down the leading coefficient to the bottom row.
• Multiply this number by \(c\) and write the result under the next coefficient.
• Add the column and write the result in the bottom row.
• Repeat the multiply and add process until completion.

This method not only identifies zeros but also aids in finding the remainder and the quotient polynomial, the latter of which can be further analyzed for additional roots.

Rational Zeros

Rational zeros are the solutions of the polynomial function that can be expressed as a fraction or integer. To find rational zeros, we often employ the Rational Root Theorem. This theorem provides insights by suggesting all possible rational zeros based on the factors of the constant term and the leading coefficient.
During this exercise:

• The potential rational zeros were determined from the coefficients: \(\pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}\).
• Each potential zero is tested by substitution into the polynomial function to see if the function evaluates to zero.

After identifying a valid zero, further exploration is needed either through additional testing or division methods to find all zeros of the polynomial.

Factorization

Factorization is the process of breaking down a polynomial into simpler terms, or factors, that when multiplied together give the original polynomial. This technique is extremely helpful in solving polynomial equations because it simplifies complex problems into more manageable ones.
The factorization process often begins after identifying a zero of the polynomial through rational zeros or synthetic division. For example, once a zero is found like \(x = -\frac{1}{2}\), the polynomial can be divided by \(x + \frac{1}{2}\) to find more factors.

• Identify zeros or use division methods to obtain a simpler polynomial (the quotient).
• Continue factorization until all zeros are found or the polynomial is expressed as a product of lower-degree polynomials.

This method is a powerful tool in algebra and can greatly simplify solving equations and understanding the behavior of polynomial functions.