Problem 434 - Precalculus

Question

List all possible rational zeros for the functions. $$f(x)=4 x^{5}-10 x^{4}+8 x^{3}+x^{2}-8$$

Step-by-Step Solution

Answer

The possible rational zeros are \( \pm 1, \pm \frac{1}{4}, \pm \frac{1}{2}, \pm 2, \pm 4, \pm 8 \).

Step 1: Identify the coefficients

Identify the coefficients of the polynomial function. Here, the polynomial is \(f(x) = 4x^5 - 10x^4 + 8x^3 + x^2 - 8\), so we have the coefficients for each term: \[a_5=4,\ a_4=-10,\ a_3=8,\ a_2=1,\ a_1=0\ (since\ there\ is\ no\ x\ term),\ and\ a_0=-8.\]

Step 2: Determine possible numerators (Factors of Constant Term)

List all the factors of the constant term of the polynomial, which is \(-8\). The factors of \(-8\) are \( \ \{ \pm 1, \pm 2, \pm 4, \pm 8 \} \).

Step 3: Determine possible denominators (Factors of Leading Coefficient)

List all the factors of the leading coefficient of the polynomial, which is \(4\). The factors of \(4\) are \( \ \{ \pm 1, \pm 2, \pm 4 \} \).

Step 4: Formulate the possible rational zeros

Use the Rational Root Theorem to generate the list of possible rational zeros. These are of the form \( \frac{p}{q} \) where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. Thus, we form the set: \[ \ \left\{ \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 2, \pm \frac{2}{4} \ (which\ simplifies\ to\ \pm \frac{1}{2}), \pm 4, \pm \frac{4}{2} \ (which\ simplifies\ to\ \pm 2), \pm \frac{4}{4} \ (which\ simplifies\ to\ \pm 1), \pm 8, \pm \frac{8}{2} \ (which\ simplifies\ to\ \pm 4), \pm \frac{8}{4} \ (which\ simplifies\ to\ \pm 2), \pm \frac{-8}{4} \ (which\ simplifies\ to\ \pm 2) \right\} \] \[ \ = \left\{ \pm 1, \pm \frac{1}{4}, \pm \frac{1}{2}, \pm 2, \pm 4, \pm 8 \right\} \]

Step 5: List the possible rational zeros

The potential rational zeros for the polynomial \(f(x)\) are: \( \pm 1, \pm \frac{1}{4}, \pm \frac{1}{2}, \pm 2, \pm 4, \pm 8 \).

Key Concepts

Polynomial Coefficients

When dealing with polynomials, coefficients are an integral part. They are the numbers or constants that multiply the variable terms within a polynomial. For the polynomial function \[f(x) = 4x^5 - 10x^4 + 8x^3 + x^2 - 8\]we can identify its coefficients from each term. The coefficient of \(x^5\) is 4, for \(x^4\) it is -10, for \(x^3\) it is 8, for \(x^2\) it is 1, and since there's no \(x^1\) term explicitly shown, its coefficient is 0. Lastly, the constant term or \(x^0\) term is -8.

• Leading Coefficient: The coefficient at the highest degree of the polynomial, here it is 4 for the term \(4x^5\).
• Understanding Coefficients: These values directly impact the shape and position of the polynomial's graph.

These coefficients are pivotal when determining factors for the rational root theorem, helping us to explore the characteristics and behavior of polynomials.

Constant Term Factors

The constant term in a polynomial is the standalone number, or the value of the polynomial when \(x\) is zero. In the polynomial \[f(x) = 4x^5 - 10x^4 + 8x^3 + x^2 - 8\]the constant term is -8. To use the Rational Root Theorem, knowing the factors of the constant term is crucial as they serve as possible numerators for potential rational roots.

• Factors of the Constant Term: The factors of -8 are \(\pm 1, \pm 2, \pm 4, \pm 8\).
• Role in Rational Root Theorem: They provide potential values for \(p\) in the \(\frac{p}{q}\) form used to determine rational zeros.

Identifying these factors is a key step in predicting which rational numbers might be roots of the polynomial function.

Leading Coefficient Factors

In polynomial functions, the leading coefficient is the coefficient of the term with the highest degree. For \[f(x) = 4x^5 - 10x^4 + 8x^3 + x^2 - 8\]the leading coefficient is 4, which is associated with \(x^5\). Discovering the factors of this leading coefficient is significant when applying the Rational Root Theorem.

• Factors of the Leading Coefficient: The factors of 4 are \(\pm 1, \pm 2, \pm 4\).
• Use in Rational Root Theorem: They provide possible denominators for the potential rational zeros, represented as \(\frac{p}{q}\).

These denominators, combined with the factors of the constant term, help in establishing a list of plausible rational zeros for the polynomial.

Possible Rational Zeros

The Rational Root Theorem is a powerful tool that helps us predict possible rational zeros of polynomial functions. It states that any possible rational zero of a polynomial, with integer coefficients, is of the form \(\frac{p}{q}\), where \(p\) and \(q\) are factors of the constant term and leading coefficient, respectively.
Given the polynomial \[f(x) = 4x^5 - 10x^4 + 8x^3 + x^2 - 8\],we use:

• The factors of the constant term (-8): \(\pm 1, \pm 2, \pm 4, \pm 8\).
• The factors of the leading coefficient (4): \(\pm 1, \pm 2, \pm 4\).

By forming the ratios \(\frac{p}{q}\) and simplifying, we get the list of potential rational zeros:\(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 2, \pm 4, \pm 8\).
Each value represents a potential x-intercept of the polynomial function's graph, aiding in understanding where the graph might cross the x-axis.