Problem 5 - Precalculus

Question

List the possibilities for rational roots. $$4 x^{3}-9 x^{2}-15 x+3=0$$

Step-by-Step Solution

Answer

Possible rational roots: \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}\).

Step 1: Identify Coefficients and Constant Terms

In the polynomial \(4x^3 - 9x^2 - 15x + 3 = 0\), identify the leading coefficient \(a\) and the constant term \(b\). Here, \(a = 4\) and \(b = 3\).

Step 2: List Factors of the Constant Term

Factors of the constant term \(3\) are \(\pm 1\) and \(\pm 3\). List these as possible numerators for rational roots.

Step 3: List Factors of the Leading Coefficient

Factors of the leading coefficient \(4\) are \(\pm 1\), \(\pm 2\), and \(\pm 4\). List these as possible denominators for rational roots.

Step 4: Form Possible Rational Roots

Using the Rational Root Theorem, possible rational roots are formed by the fraction of factors of the constant term over factors of the leading coefficient. Combine them to get: \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}\).

Key Concepts

Factors of Constant Term

When dealing with polynomials, particularly in the context of the Rational Root Theorem, it's important to start by identifying the factors of the constant term. In our example polynomial, \(4x^3 - 9x^2 - 15x + 3 = 0\), the constant term is \(3\). The factors of a number are the integers that can divide it without leaving a remainder.

For \(3\), the factors are simple:

• \(+1\)
• \(-1\)
• \(+3\)
• \(-3\)

These factors are critical because they will be used as the numerators when determining the possible rational roots of the polynomial. This step is fundamental in narrowing down the potential solutions that need to be tested in the polynomial equation.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It's usually represented by the letter \(a\) in expressions like \(ax^n + bx^{n-1} + ...\).

In the polynomial equation \(4x^3 - 9x^2 - 15x + 3 = 0\), the leading term is \(4x^3\). Thus, the leading coefficient is \(4\). Why is this significant? Because the factors of the leading coefficient serve as the denominators for the possible rational roots. Let's consider the factors of \(4\):

• \(+1\)
• \(-1\)
• \(+2\)
• \(-2\)
• \(+4\)
• \(-4\)

By identifying these factors, we are ready to form possible fractions that can be solutions to the equation using the Rational Root Theorem.

Possible Rational Roots

With the factors of both the constant term and the leading coefficient in hand, we can use the Rational Root Theorem to determine the possible rational roots of the polynomial. The theorem states that any possible rational root, expressed as a fraction \(\frac{p}{q}\), must have:

• \(p\) as a factor of the constant term, \(3\)
• \(q\) as a factor of the leading coefficient, \(4\)

By combining the factors identified in the previous sections, we form these possible rational roots:

• \(+1\)
• \(-1\)
• \(\frac{1}{2}\)
• \(-\frac{1}{2}\)
• \(\frac{1}{4}\)
• \(-\frac{1}{4}\)
• \(+3\)
• \(-3\)
• \(\frac{3}{2}\)
• \(-\frac{3}{2}\)
• \(\frac{3}{4}\)
• \(-\frac{3}{4}\)

These fractions are the candidates you would test in the polynomial equation to discover any valid rational roots. By systematically trying these values, you can determine which, if any, are actual solutions to the polynomial equation.