Problem 73 - College Algebra

Question

The equations in Exercises \(72-75\) have real roots that are rational. Use the Rational Zero Theorem to list all possible rational roots. Then graph the polynomial function in the given viewing rectangle to determine which possible rational roots are actual roots of the equation. $$ 6 x^{3}-19 x^{2}+16 x-4-0 ;[0,2,1] \text { by }[-3,2,1] $$

Step-by-Step Solution

Answer

Applying the Rational Zero Theorem allows you to list possible rational roots. The actual roots are then determined by plotting the polynomial and identifying where it intersects the x-axis.

Step 1: Applying the Rational Zero Theorem

List all the factors of the constant -4 and the leading coefficient 6. The factors of -4 are ±1, ±2, ±4, and the factors of 6 are ±1, ±2, ±3, ±6. Use these to form the \(p/q\) fractions, which are our possible rational roots.

Step 2: Forming the Possible Roots

The possible rational roots are ±1/1, ±2/1, ±4/1, ±1/2, ±2/2, ±4/2, ±1/3, ±2/3, ±4/3, ±1/6, ±2/6, and ±4/6. Simplifying these gives ±1, ±2, ±4, ±1/2, ±2, ±1, ±1/3, ±2/3, ±4/3, ±1/6, ±1/3, and ±2/3.

Step 3: Graphing the Polynomial

Plot the polynomial equation \(6x^3 - 19x^2 + 16x - 4 = 0\) using a graphing calculator or software within the viewing rectangle [0,2,1] by [-3,2,1]. Observe where the graph intersects or touches the x-axis within the specified viewing rectangle, as those x-values are the actual roots.

Step 4: Identifying the Real Roots

From the graph, identify which of the possible roots from Step 2 are actual roots of the equation based on where the curve intersects the x-axis. Those are the actual rational roots.

Key Concepts

Understanding Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables with non-negative integer exponents and coefficients. The general form of a polynomial function can be expressed as
\( f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_2x^2 + a_1x + a_0 \),
where \(n\) is a non-negative integer, \(a_n\) is the leading coefficient, and \(a_0\) is the constant term. The degree of the polynomial is the highest power of \(x\) that appears in the function.

In the exercise, we are dealing with a cubic polynomial function, as the highest power of \(x\) is three. Understanding the structure of polynomial functions is crucial to applying the Rational Zero Theorem effectively, as it helps us determine the possible rational roots by examining the coefficients of the highest and lowest degree terms.

The Rational Zero Theorem Explained

The Rational Zero Theorem is a vital tool in finding rational solutions to polynomial equations. This theorem states that if a polynomial equation with integer coefficients has any rational roots, they must be a fraction formed by a factor of the constant term over a factor of the leading coefficient.

In layman's terms, if you have a list of all the factors of the last number in the polynomial (the constant term) and the first number (the leading coefficient), each factor of the constant term can be divided by each factor of the leading coefficient to get all possible rational roots. Constructing this list of potential roots enables a systematic approach to narrow down which are the actual roots.

In the textbook exercise, you start with the constant term \( -4 \) and the leading coefficient \( 6 \). By listing these factors and creating rational fractions, you get possible roots like \( \pm1 \) and \( \pm\frac{1}{2} \). These possible roots are pivotal for identifying which ones are actual solutions to the polynomial equation.

Graphing Polynomials to Determine Roots

Graphing polynomials can provide a visual representation to better understand where the function's roots lie, which are the values of \( x \) for which \( f(x) = 0 \). The x-intercepts of the graph, where the curve crosses the x-axis, correspond to the roots.

Using graphing tools, like calculators or software, we can plot the given polynomial function within specified intervals. For instance, the viewing rectangle [0,2,1] by [-3,2,1] gives a focused window to accurately detect the points where the function touches or crosses the x-axis.

After plotting, you examine the points of intersection with the x-axis. Cross-referencing these x-values with the list of possible rational roots obtained using the Rational Zero Theorem then solidifies which are the actual rational roots of the equation. Learning to graph polynomials not only aids in finding roots but also in understanding the overall behavior of polynomial functions.