Problem 8 - College Algebra

Question

In Exercises \(1-8,\) use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)-4 x^{5}-8 x^{4}-x+2 $$

Step-by-Step Solution

Answer

The possible rational zeros for the given function are \( \pm 1, \pm 0.5, \pm 0.25, \pm 2 \).

Step 1: Identify Leading Coefficient and Constant term

For \( f(x) = -4x^{5} -8x^{4} -x + 2 \), the constant term is \( 2 \) and the leading coefficient is \( -4 \). All possible values of \( p \) are factors of \( 2 \) and all possible values of \( q \) are factors of \( -4 \).

Step 2: Determine Factors

The factors of the constant term \( 2 \) are \( \pm 1, \pm 2 \). The factors of the leading coefficient \( -4 \) are \( \pm 1, \pm 2, \pm 4 \).

Step 3: List the Rational Zeros

Generate a list of rational zeros by putting every possible \( p \) over every possible \( q \), and simplifying the fraction. Don't forget both the positive and negative possibilities. The rational zeros can thus be \( \pm \frac{1}{1}, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{2}{1}, \pm \frac{2}{2}, \pm \frac{2}{4} \). After simplifying, the possible rational roots are \( \pm 1, \pm 0.5, \pm 0.25, \pm 2, \pm 0.5, \pm 1 \)

Step 4: Remove Duplicates

Remove duplicate zeros from the list. The final list of possible rational zeros for \( f(x) = -4x^{5} -8x^{4} -x + 2 \) is \( \pm 1, \pm 0.5, \pm 0.25, \pm 2 \)

Key Concepts

Leading Coefficient

In the polynomial equation provided, the leading coefficient is extremely important. It is the coefficient of the term with the highest power of the variable. In this case, the equation is \( f(x) = -4x^5 -8x^4 - x + 2 \). Here, \(-4\) is the leading coefficient because \(-4x^5\) is the term with the highest degree, i.e., the highest power of \(x\). The leading coefficient helps to determine the nature of the polynomial's graph.
The sign of the leading coefficient influences the end behavior of the polynomial's graph. Also, in using the Rational Zero Theorem to find possible rational zeros, we need the factors of the leading coefficient. Knowing these factors is crucial because they form the denominator of possible rational zeros in the form \(\frac{p}{q}\).

• Tip: Always look for the term with the highest \(x\) exponent to find the leading coefficient.
• Reminder: The leading coefficient is essential for graphing polynomials and calculating rational zeros.

Constant Term

In a polynomial equation, the constant term is the term without any variables—it stands alone. For the equation \( f(x) = -4x^5 -8x^4 - x + 2 \), the constant term is \(2\). This is because it does not have an \(x\) attached to it.
The constant term is important when applying the Rational Zero Theorem, as it helps us generate possible values of \(p\) in the rational zero formula \(\frac{p}{q}\). Here, the factors of \(2\) are \(\pm 1\) and \(\pm 2\).

• Understanding the constant term: It's simply the number all by itself in the equation.
• Use: It helps to determine potential numerators for rational zeros.
• Quick Reminder: Always check for the standalone number as your constant term.

Factors

When we talk about factors in the context of the Rational Zero Theorem, we mean breaking down numbers into their components that, when multiplied together, give the original number. Let's see how this works for the equation
\[ f(x) = -4x^5 -8x^4 - x + 2 \].
Factors of the constant term \(2\) are \(\pm 1, \pm 2\), and factors of the leading coefficient \(-4\) are \(\pm 1, \pm 2, \pm 4\).
These factors are crucial because, in the theorem, we express potential rational zeros as \(\frac{p}{q}\), where:

• \(p\) comes from the factors of the constant term.
• \(q\) comes from the factors of the leading coefficient.

This method systematically generates a list of potential rational zeros, allowing us to explore solutions more efficiently. By simplifying \(\frac{p}{q}\), we find all potential zeros, improving our equation-solving strategies. It is a great way to narrow down potential zeros before deciding which ones to test thoroughly.