Problem 57 - College Algebra

Question

The equations 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

The potential rational roots of the polynomial obtained from the Rational Root Theorem included ±1, ±2, ±4, ±1/2, ±1/3, ±2/3, ±1/6. After graphing it, the actual roots found within the viewing rectangle were x = 1 and x = 2/3.

Step 1: Applying Rational Zero Theorem

The Rational Zero Theorem says that any rational root of a polynomial equation \(ax^n + bx^{n-1} + ... k = 0\) can be expressed as \(p/q\) where p is a factor of the constant term (in this case -4) and q is a factor of the leading coefficient (in this case 6). Factor 4 and 6. Factors of 4: ±1, ±2, ±4. Factors of 6: ±1, ±2, ±3, ±6. So all 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, ±4/6. Simplifying gives the following list: ±1, ±2, ±4, ±1/2, ±1/3, ±2/3, ±1/6.

Step 2: Graphing the Function

For the graphing part, one can use any graphing tool or calculator. The viewing rectangle given is [0,2,1] by [-3,2,1] which specifies the range of x-values and y-values to view. By graphing the polynomial and observing the x-intercepts within this range, the actual roots can be found.

Step 3: Identifying the Roots

After graphing the function, one can observe where the function intersects the x-axis within the viewing rectangle. These points are the roots of the polynomial. The x-intercepts seem to coincide with x = 1 and x = 2/3 from the possible roots derived from the Rational Root Theorem.

Key Concepts

Polynomial Equations

Polynomial equations are equations that involve a polynomial expression. In mathematical terms, a polynomial is a sum of various terms, each consisting of a variable raised to an integer power and multiplied by a coefficient. These equations can look complex, but they are essential because they appear frequently in various fields of mathematics and science. Here's what you need to remember:

• The degree of the polynomial depends on the highest power of the variable, known as the degree of the polynomial. For example, in the equation \(6x^3 - 19x^2 + 16x - 4 = 0\), the highest power is 3, making it a cubic polynomial.
• Polynomials are generally expressed in standard form, which means they are ordered from the highest degree to the lowest degree.
• Solving polynomial equations often involves finding the roots or zeros of the polynomial. These are the x-values where the polynomial evaluates to zero.

Understanding polynomial equations is a foundational skill in algebra that can lead to solving more intricate problems. They can be tackled using various methods, such as factoring, graphing, or applying the Rational Root Theorem.

Graphing Polynomial Functions

Graphing polynomial functions is an excellent way to visualize the roots and behavior of these functions. When you graph a polynomial equation like \(6x^3 - 19x^2 + 16x - 4 = 0\), you are essentially mapping out all its potential roots and where it intersects the x-axis. To graph effectively:

• Identify the range of x-values and y-values, which helps focus on the interesting parts of the graph. In our problem, we're looking at \([0,2]\) for x and \([-3,2]\) for y.
• A graphing calculator or software can be used for more accurate and easier visualization.
• The x-intercepts represent the real roots of the polynomial. Pay attention to where the curve crosses the x-axis, as these points mark the roots.

Graphing helps in assessing whether the roots are simple (crossing the x-axis) or repeated (just touching it). Additionally, it allows us to check which roots from the Rational Root Theorem are actual zeros of the polynomial function.

Real and Rational Roots

Real and rational roots are critical concepts when dealing with polynomial equations. A root of a polynomial is the value of the variable that makes the polynomial equal zero. Here's what distinguishes them:

• Real Roots: These are the x-intercepts on the graph of the polynomial where the polynomial equals zero. In the given polynomial equation, these roots are evident where the curve meets the x-axis.
• Rational Roots: These are specifically the roots that can be expressed as a fraction \(\frac{p}{q}\). The Rational Root Theorem provides a way to list all possible rational roots by examining the factors of the constant term and the leading coefficient.

In our exercise, the Rational Root Theorem helps predict potential rational roots, but not all listed will work. By graphing the equation, we can confirm which ones are actual rational roots, such as \(x = 1\) and \(x = \frac{2}{3}\), the points where the polynomial crosses the x-axis.