Problem 89 - Precalculus Mathematics for Calculus

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. $$x^{3}-3 x^{2}-4 x+12=0 ; \quad[-4,4] \text { by }[-15,15]$$

Step-by-Step Solution

Answer

The actual rational roots are \( x = 2 \) and \( x = -2 \).

Step 1: Understanding the Polynomial and the Rational Root Theorem

The given polynomial is a cubic equation: \( x^3 - 3x^2 - 4x + 12 = 0 \). The Rational Root Theorem helps us identify possible rational roots by taking all factors of the constant term (12) and dividing them by all factors of the leading coefficient (1, because this is a monic polynomial).

Step 2: Identify Possible Rational Roots

The constant term is 12, so the possible factors are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \). As the leading coefficient is 1, the rational root candidates remain \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).

Step 3: Graph the Polynomial

Plot the polynomial \( x^3 - 3x^2 - 4x + 12 \) within the viewing window of \([-4,4]\) for the x-axis and \([-15,15]\) for the y-axis. Look for points where the graph crosses the x-axis, as these are the roots.

Step 4: Determine Actual Rational Roots from the Graph

Upon graphing, determine where the curve intersects the x-axis at integer points. Visual inspection from the graph shows that the roots are at \( x = 2 \) and \( x = -2 \). Evaluate the polynomial at these points to verify.

Step 5: Verify Solutions by Substitution

Substitute \( x = 2 \) into the polynomial: \( 2^3 - 3\times2^2 - 4\times2 + 12 = 0 \) and \( x = -2 \): \( (-2)^3 - 3(-2)^2 - 4(-2) + 12 = 0 \). Both calculations equal zero, confirming these are actual roots.

Key Concepts

Polynomial Graphing

Graphing polynomials helps visualize where the function crosses the x-axis. These crossing points indicate the roots of the polynomial. When graphing the polynomial \( x^3 - 3x^2 - 4x + 12 \) over the window \([-4, 4]\) for the x-axis and \([-15, 15]\) for the y-axis, we're particularly interested in where the graph reaches zero on the y-axis. This happens at the x-values of the roots. By plotting the curve, we can determine these specific points.

To graph efficiently:

• Understand the polynomial's degree and behavior at infinity: A cubic polynomial will have end behaviors like \( x^3 \), going to positive infinity in one direction and negative infinity in the other.
• Identify real roots: These are any x-values where the graph crosses the x-axis.

Graphing not only shows us the approximate location of the roots but can also help confirm calculations or predictions made using algebraic methods.

Rational Root Theorem

The Rational Root Theorem is a valuable tool in algebra, particularly for finding the potential rational roots of a polynomial equation.

For a polynomial given by \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0 \), where \( a_0 \) is the constant term and \( a_n \) is the leading coefficient, the theorem states that any rational solution, or root, must be a ratio of a factor of \( a_0 \) to a factor of \( a_n \).

In simpler terms:

• The possible rational roots are given by \( \frac{p}{q} \), where \( p \) is a factor of the constant term (here, 12) and \( q \) is a factor of the leading coefficient (here, 1).
• This results in a list of potential roots: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).

This list provides a guide to test which values are actual roots. To find out the true roots, one can substitute these values into the polynomial and check if the result is zero, or verify using graphing.

Cubic Polynomials

Cubic polynomials are equations of the form \( ax^3 + bx^2 + cx + d \). These polynomials can have up to three roots, and the shape of their graphs is uniquely characterized by a single turning point or a set of inflection points.

Key characteristics of cubic polynomials include:

• They can intersect the x-axis at up to three points, reflecting up to three real roots, including repeated roots.
• The equation \( x^3 - 3x^2 - 4x + 12 \) has a leading term \( x^3 \), indicating its degree is 3, which implies it can have at most three roots.
• The graph of a cubic polynomial may appear as a smooth curve that can twist and turn, depending on the coefficients.

When solving cubic polynomials, it's essential to identify multiple potential real roots, both through algebraic methods and graphing. Moreover, understanding the turning and inflection points aids in predicting the graph's shape and behavior.