Problem 3 - College Algebra

Question

In Exercises \(1-8,\) use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)-3 x^{4}-11 x^{3}-x^{2}+19 x+6 $$

Step-by-Step Solution

Answer

The potential rational zeros for the polynomial equation are ±1, ±2, ±3, ±6, ±1/3, ±2/3.

Step 1: Identify the values of a and d

From here, we can see that our \(a\) value (leading coefficient) is 3, and our \(d\) value is 6 (constant term).

Step 2: Find all of the possible values of p and q

All values of \(p\) are numbers that divide evenly into our \(d\) value. In this case, \(d = 6\), so the possible values of \(p\) are ±1, ±2, ±3, ±6. Similarly, our values of \(q\) are numbers that divide evenly into our \(a\) value, 3. Therefore, the possible values for \(q\) are ±1, ±3.

Step 3: List all possible rational zeros

Finally, we list all the possible rational zeros of the polynomial \(f(x)\), which will be all fractions in the form \(\frac{p}{q}\) where \(p\) is a factor of 6 and \(q\) is a factor of 3. Using the combination of values for \(p\) and \(q\) derived in Step 2, we then look for unique divisions. Listing all possible rational zeros, we can arrive at the following set: ±1, ±2, ±3, ±6, ±1/3, ±2/3.

Key Concepts

Polynomials

Polynomials are mathematical expressions that consist of variables and coefficients, constructed using only addition, subtraction, multiplication, and non-negative integer exponents of variables. They are foundational in algebra and are often expressed in the form:

• The general form of a polynomial is: \[ a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \]
• Where each \(a_i\) represents a constant coefficient, and \(x\) is a variable.
• The degree of the polynomial is determined by the highest exponent.

Polynomials can be categorized by the number of terms they have, such as monomials (one term), binomials (two terms), or trinomials (three terms). In our exercise, the polynomial \(3x^4-11x^3-x^2+19x+6\) is of degree 4, thus making it a quartic polynomial. Understanding polynomials is essential to mastering the Rational Zero Theorem, as you need to know how to identify different parts to find rational zeros.

Rational Zeros

Rational zeros of a polynomial are values for which the polynomial evaluates to zero and can be expressed as a fraction of integers. Finding rational zeros can help factor and simplify polynomials, making them easier to solve. The Rational Zero Theorem provides a systematic way to determine these zeros.

• A rational zero is any zero of a polynomial that can be written as \(\frac{p}{q}\), where \(p\) and \(q\) are integers.
• To find potential rational zeros, consider the factors of the constant term and leading coefficient.

Listing all combinations of \(\frac{p}{q}\) where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient gives potential rational zeros. This method does not guarantee that all these values will be zeros, but it identifies candidates for verification via substitution into the polynomial.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. This coefficient plays an important role in determining the polynomial's terms and behavior as \(x\) approaches infinity. In the context of the Rational Zero Theorem, the leading coefficient is also essential.

• It determines the possible values of \(q\) in potential rational zeros \(\frac{p}{q}\).
• It influences the end behavior of the polynomial function graph.
• In our exercise, the leading coefficient is 3, which means potential values for \(q\) are ±1 and ±3.

By understanding the significance of the leading coefficient, students can more effectively apply the Rational Zero Theorem and predict polynomial behavior in various contexts.

Constant Term

The constant term in a polynomial is the term with no variable attached, represented as \(a_0\) in the general form. It is critical when applying the Rational Zero Theorem to find rational zeros. For our polynomial \(f(x)=3x^4-11x^3-x^2+19x+6\), the constant term is 6.

• The constant term \(d\) provides the possible values of \(p\) for rational zeros \(\frac{p}{q}\).
• Potential values of \(p\) for our example are ±1, ±2, ±3, ±6.
• The constant term does not affect the polynomial's degree or highest power but is integral to determining its root.

Recognizing the role of the constant term enables students to successfully derive and test potential rational zeros, streamlining the process of polynomial evaluation and solution.