Problem 84 - Algebra and Trigonometry

Question

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$ x^{3}-3 x^{2}-4 x+12=0 ;[-4,4] \text { by }[-15,15] $$

Step-by-Step Solution

Answer

The rational solution of the polynomial is \(x = 2\).

Step 1: Understanding the Rational Zeros Theorem

The Rational Zeros Theorem states that any rational solution of a polynomial equation of the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0\) is of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading coefficient \(a_n\). For the polynomial \(x^3 - 3x^2 - 4x + 12\), \(a_0 = 12\) and \(a_n = 1\).

Step 2: Listing Factors of the Constant Term and Leading Coefficient

Factors of the constant term \(12\) are \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\). The only factor of the leading coefficient \(1\) is \(\pm 1\).

Step 3: Determining the Possible Rational Roots

Using the Rational Zeros Theorem, the possible rational roots are the factors of the constant term divided by the factors of the leading coefficient. This gives \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\).

Step 4: Graphing the Polynomial

Plot the polynomial \(x^3 - 3x^2 - 4x + 12\) in the viewing rectangle \([-4, 4]\) by \([-15, 15]\). Visually inspect the graph to identify where the curve crosses the x-axis to identify potential roots.

Step 5: Verifying the Solutions

From the graph, determine which potential rational roots actually cause the polynomial to equal zero. Confirm by evaluating the polynomial at these points: \(x=2\) results in \(f(2)=0\), meaning \(x=2\) is a root.

Key Concepts

Polynomial Equations

Polynomial equations are mathematical expressions involving a sum of powers of a variable, often denoted as \(x\). In standard form, a polynomial equation can be written as \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0\). Here, each term is constructed by multiplying a coefficient \(a_n, a_{n-1}, \ldots\) with a power of \(x\). The highest power determines the degree of the polynomial.

For example, the polynomial \(x^3 - 3x^2 - 4x + 12 = 0\) is a cubic equation because the highest power of \(x\) is 3. Solving polynomial equations involves finding the values of \(x\) that make the entire expression equal to zero, known as the roots or solutions of the equation. These solutions might be real or complex numbers.

Rational Roots

Rational roots, also known as rational zeros, are the solutions of a polynomial equation that can be expressed as a fraction \(\frac{p}{q}\) where both \(p\) and \(q\) are integers, and \(q eq 0\). The Rational Zeros Theorem is an essential method used to identify these potential roots.

This theorem provides a systematic way to test which of the conceivable rational numbers can actually satisfy the polynomial equation. By testing these against the polynomial, we can determine if they make the polynomial equal zero.

Factors of Constant Term

The constant term of a polynomial, denoted as \(a_0\), is the term without any variable attached. When applying the Rational Zeros Theorem, the factors of the constant term are crucial as they help identify the values for \(p\) in the fraction \(\frac{p}{q}\).

In the example \(x^3 - 3x^2 - 4x + 12 = 0\), the constant term is 12. Its factors are \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\). These factors indicate all possible numerators for the rational roots, assuming unity is the factor for the leading coefficient.

Leading Coefficient

The leading coefficient in a polynomial is the coefficient of the term with the highest degree. In a polynomial equation \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0\), \(a_n\) is the leading coefficient. It plays a significant role in finding rational roots through the Rational Zeros Theorem because it determines possible denominators for these roots.

For the polynomial \(x^3 - 3x^2 - 4x + 12\), the leading coefficient is 1. Hence, the factor of the leading coefficient is \(\pm 1\). This simplicity restricts the rational roots to integers, specifically because any integer divided by 1 is just that integer.