Question

List the potential rational zeros of each polynomial function. Do not attempt to find the zeros. $$ f(x)=6 x^{4}+2 x^{3}-x^{2}+20 $$

Short answer

Potential rational zeros: \ \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm 2, \pm \frac{2}{3}, \pm 4, \pm \frac{4}{3}, \pm 5, \pm \frac{5}{2}, \pm 10, \pm \frac{10}{3}, \pm 20 \).

Step-by-step solution

Identify the constant term and leading coefficient

The constant term (the term without an x) in the polynomial function is 20. The leading coefficient (the coefficient of the highest degree term) is 6 (from the term 6x^4).

List the factors of the constant term

Find all the factors of the constant term 20. They are: ±1, ±2, ±4, ±5, ±10, ±20.

List the factors of the leading coefficient

Find all the factors of the leading coefficient 6. They are: ±1, ±2, ±3, ±6.

Form the ratios of factors

Form all possible positive and negative ratios of the factors of the constant term over the factors of the leading coefficient. This means you generate all combinations \ \(\frac{\text{constant factor}}{\text{leading coefficient factor}} \).

List the potential rational zeros

The potential rational zeros are: \ \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm 2, \pm \frac{2}{3}, \pm \frac{2}{6}, \pm 4, \pm \frac{4}{3}, \pm \frac{4}{6}, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{6}, \pm 10, \pm \frac{10}{3}, \pm \frac{10}{6}, \pm 20, \pm \frac{20}{3}, \pm \frac{20}{6} \). Simplify where possible to obtain a final list.

Key concepts

constant term

In a polynomial, the constant term is the term without any variables attached to it. It's just a number by itself. In our given exercise, the polynomial is \(f(x) = 6x^4 + 2x^3 - x^2 + 20\). Here, the constant term is the number 20. This term is critical when we apply the *Rational Zeros Theorem*. Identifying the constant term early on will help us list all possible rational roots later.
Remember, the constant term can be thought of as the standalone 'y-intercept' if you were to graph the polynomial. It gives valuable information about the polynomial's behavior at \(x = 0\).

leading coefficient

The leading coefficient is the coefficient of the term with the highest degree in the polynomial. In the polynomial \(f(x) = 6x^4 + 2x^3 - x^2 + 20\), the term with the highest degree is \(6x^4\), so the leading coefficient is 6.
The leading coefficient plays an essential role in determining the possible rational zeros of the polynomial. When combined with the constant term's factors, it helps us form potential zeros through ratios.
In simpler terms, the leading coefficient tells us about the polynomial's 'steepness.' A higher leading coefficient generally means that the polynomial's graph rises or falls more steeply as \(x\) moves away from zero.

rational zeros theorem

The Rational Zeros Theorem is a powerful tool for finding potential rational zeros (roots) of a polynomial. According to this theorem, any rational solution of the polynomial equation \(f(x) = 0\) is a fraction \( \frac{p}{q} \), where:

• *\(p\)* is a factor of the constant term
• *\(q\)* is a factor of the leading coefficient

For our polynomial \(f(x) = 6x^4 + 2x^3 - x^2 + 20\), the factors of the constant term (20) are ±1, ±2, ±4, ±5, ±10, and ±20. The factors of the leading coefficient (6) are ±1, ±2, ±3, and ±6.
Combining these factors results in all possible rational roots. We find these by forming ratios like \(\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \dots\), including both positive and negative values, leading to our final set of potential zeros.

factorization

Factorization refers to breaking down a polynomial into its simplest building blocks (factors) that, when multiplied, give back the polynomial. For instance, if we can express a polynomial as the product of linear factors like \((x - a)(x - b)(x - c)\), then a, b, and c are the zeros of the polynomial.
Factorizing polynomials can sometimes be complex and may require several techniques like grouping, synthetic division, or the Rational Zeros Theorem itself. The primary goal of factorization is to simplify the polynomial to find its roots easily.
In our exercise, although we are not asked to factorize completely, understanding factorization is essential. Knowing how to factor can simplify identifying zeros and rooting out the rational solutions found using the Rational Zeros Theorem.