Problem 28 - Algebra 2

Question

Use the Rational Root Theorem to list all possible rational roots for each polynomial equation. Then find any actual rational roots. $$ 8 x^{3}-28 x^{2}+14 x+15=0 $$

Step-by-Step Solution

Answer

The possible rational roots are \rmpm1, \rmpm3, \rmpm5, \rmpm15, \rmpm\frac{1}{2}, \rmpm\frac{3}{2}, \rmpm\frac{5}{2}, \rmpm\frac{15}{2}, \rmpm\frac{1}{4}, \rmpm\frac{3}{4}, \rmpm\frac{5}{4}, \rmpm\frac{15}{4}, \rmpm\frac{1}{8}, \rmpm\frac{3}{8}, \rmpm\frac{5}{8}, \rmpm\frac{15}{8}. Test each possible root to find the actual rational roots.

Step 1: List all possible rational roots using the Rational Root Theorem

The Rational Root Theorem states that any rational root for the polynomial equation, which is of the form \( p(x) = a_nx^n + \dots + a_1x + a_0 \), must be of the form \(\frac{p}{q}\), where \(p\) divides the constant term \(a_0\) and \(q\) divides the leading coefficient \(a_n\). For the given polynomial \(8x^3 - 28x^2 + 14x + 15 = 0\), the constant term is 15 and the leading coefficient is 8. The divisors of 15 are \(\rmpm 1, \rmpm 3, \rmpm 5, \rmpm 15\), and the divisors of 8 are \(\rmpm 1, \rmpm 2, \rmpm 4, \rmpm 8\). Thus, the possible rational roots are \( \rmpm\frac{1}{1}, \rmpm\frac{3}{1}, \rmpm\frac{5}{1}, \rmpm\frac{15}{1}, \rmpm\frac{1}{2}, \rmpm\frac{3}{2}, \rmpm\frac{5}{2}, \rmpm\frac{15}{2}, \rmpm\frac{1}{4}, \rmpm\frac{3}{4}, \rmpm\frac{5}{4}, \rmpm\frac{15}{4}, \rmpm\frac{1}{8}, \rmpm\frac{3}{8}, \rmpm\frac{5}{8}, \rmpm\frac{15}{8}\).

Step 2: Simplify the list of possible rational roots

Simplify the fractions in the list of possible rational roots to obtain the unique values. The simplified list is \( \rmpm1, \rmpm3, \rmpm5, \rmpm15, \rmpm\frac{1}{2}, \rmpm\frac{3}{2}, \rmpm\frac{5}{2}, \rmpm\frac{15}{2}, \rmpm\frac{1}{4}, \rmpm\frac{3}{4}, \rmpm\frac{5}{4}, \rmpm\frac{15}{4}, \rmpm\frac{1}{8}, \rmpm\frac{3}{8}, \rmpm\frac{5}{8}, \rmpm\frac{15}{8}\). Note that some of the fractions may be equivalent to the whole numbers or each other after simplification.

Step 3: Test each possible rational root

Plug in each possible rational root into the polynomial \(8x^3 - 28x^2 + 14x + 15\) to determine if it is indeed a root. The actual rational roots are those for which the polynomial evaluates to zero.

Step 4: Find the actual rational roots

Upon testing each candidate from the list in the polynomial equation, the actual rational roots can be identified. If \(x = r\) is an actual root, then \(8r^3 - 28r^2 + 14r + 15 = 0\).

Key Concepts

Polynomial Equations

Polynomial equations are mathematical expressions that equate a polynomial to zero. A polynomial is a sum of terms, each consisting of a variable raised to a non-negative integer power and multiplied by a coefficient. These equations are foundational in algebra and are used to describe various mathematical and real-world phenomena.

For instance, the polynomial equation given in the exercise, \(8x^3 - 28x^2 + 14x + 15 = 0\), is a third-degree polynomial equation because the highest power of the variable, \(x\), is three. Solving polynomial equations usually involves finding the values of the variable that make the equation true. These values are known as the roots or zeroes of the polynomial.

Possible Rational Roots

The Rational Root Theorem provides a way to list all possible rational roots of a polynomial equation. A rational root is expressed as a fraction, \(\frac{p}{q}\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. For the polynomial \(8x^3 - 28x^2 + 14x + 15\), the constant term is 15 and the leading coefficient is 8. Following this theorem, we examine all combinations of the divisors of 15 and 8.

This process helps narrow down the potentially infinite number of rational numbers to a finite set of candidates for roots, making the task of finding actual roots more manageable.

Simplifying Fractions

Simplifying fractions is a common method in mathematics to reduce fractions to their simplest form, where the numerator and the denominator have no common factors other than 1. This step is crucial when working with the Rational Root Theorem, as it helps eliminate duplicate roots that appear in different fractional forms.

For example, the fraction \(\frac{15}{2}\) cannot be simplified further, but \(\frac{6}{8}\) simplifies to \(\frac{3}{4}\) after dividing both the numerator and the denominator by their greatest common divisor, which is 2. Simplification is essential for making the list of potential rational roots more concise and easier to test.

Testing Polynomial Roots

Testing polynomial roots involves substituting each possible rational root from the list generated by the Rational Root Theorem into the polynomial equation. If the substitution yields a result of zero, then the tested number is an actual root of the polynomial.

For our polynomial \(8x^3 - 28x^2 + 14x + 15\), we would substitute each potential root into \(x\) and evaluate the outcome. Only those values that satisfy the equation are considered actual roots. This process of test and evaluation is a surefire way of verifying which of the possible roots are indeed true solutions to the polynomial equation.