Problem 5 - College Algebra

Question

In Exercises \(1-8,\) use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)-4 x^{4}-x^{3}+5 x^{2}-2 x-6 $$

Step-by-Step Solution

Answer

The possible rational zeros of the function \(f(x)=4x^{4}-x^{3}+5x^{2}-2x-6\) are: \(\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{1}{4}, and \pm\frac{3}{4}\)

Step 1: List all Factors of the Constant and the Leading Coefficient

First, list out all the factors of the constant -6 and the leading coefficient 4. The factors of -6 are: \( \pm1, \pm2, \pm3, \pm6 \); and The factors of 4 are: \( \pm1, \pm2, \pm4 \).

Step 2: Generate Possible Rational Zeros

Now that the factors of -6 and 4 are enlisted, generate a list of the possible rational zeros using the formula \(\frac{p}{q}\), where \(p\) is a factor of -6 and \(q\) is a factor of 4. These would be:\(\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{1}{4}, \pm\frac{3}{4}, \pm\frac{2}{4}, \pm \frac{1}{4}\) Simplify the fractions to get the final list: \(\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{1}{4}, \pm\frac{3}{4}, \pm\frac{1}{2}, \pm \frac{1}{4}\)

Key Concepts

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to whole number powers, where these powers are known as the degree of the polynomial. For example, the polynomial function given in the exercise is:\[ f(x) = -4x^4 - x^3 + 5x^2 - 2x - 6 \]The degree of the polynomial is the highest power of the variable, which in this case is 4. The term with this highest power is called the leading term and in this polynomial, it is \[-4x^4\]Polynomial functions are continuous and smooth curves when graphed. They can have multiple roots or zeros, which are the values of the variable that make the polynomial equal to zero. These roots can be rational or irrational, and their identification is essential for understanding the polynomial's behavior.
Understanding polynomial functions is crucial because they appear frequently in many areas of mathematics and science, providing a model for a variety of natural phenomena.

Factors of Integers

Factors of integers are numbers that divide a given integer completely without leaving any remainder. When considering the polynomial function's constant term and its leading coefficient, identifying their factors becomes essential. This process helps in determining the possible rational roots of the polynomial.
In the exercise, the constant term is -6, which has the following factors:

• \(\pm1\)
• \(\pm2\)
• \(\pm3\)
• \(\pm6\)

Similarly, the leading coefficient of the polynomial is 4, and its factors are:

• \(\pm1\)
• \(\pm2\)
• \(\pm4\)

Finding these factors is a crucial step because they are used in the Rational Zero Theorem to find the list of possible rational roots. Factors help simplify the polynomial into manageable terms, and understanding them provides insight into polynomial behavior.

Possible Rational Roots

The Rational Zero Theorem is a valuable tool for finding possible rational roots of a polynomial function. It states that for a polynomial equation \[ a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0\]any rational solution can be expressed as \(\frac{p}{q}\), where \(p\) is a factor of the constant term \(a_0\), and \(q\) is a factor of the leading coefficient \(a_n\).
In the given exercise, the constant term is -6 and the leading coefficient is 4. By using their factors, we calculate the possible rational roots using the formula \(\frac{p}{q}\). This results in a collection of potential roots such as:

• \(\pm1\)
• \(\pm2\)
• \(\pm3\)
• \(\pm6\)
• \(\pm\frac{1}{2}\)
• \(\pm\frac{3}{2}\)
• \(\pm\frac{1}{4}\)
• \(\pm\frac{3}{4}\)

These are not guaranteed to be roots but rather potential candidates for rational solutions. Testing each one in the polynomial helps find which, if any, are actual roots of the polynomial.

Leading Coefficient

The leading coefficient of a polynomial function is the coefficient of the term with the highest degree or the largest exponent. It plays a crucial role in affecting the end behavior of the polynomial and in determining possible rational roots.
In the polynomial from the exercise:\[ f(x) = -4x^4 - x^3 + 5x^2 - 2x - 6\]The leading coefficient is -4, associated with the leading term \(-4x^4\). This coefficient affects how the polynomial grows as \(x\) becomes very large or very small, influencing the curve's stretching or compressing.
Additionally, when using the Rational Zero Theorem, the factors of the leading coefficient become crucial. They combine with the factors of the constant term to form the set of possible rational roots. Understanding the role of the leading coefficient helps predict the polynomial’s behavior and facilitates solving and factoring polynomials effectively.