Problem 58 - 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. $$ \begin{aligned}&2 x^{4}+7 x^{3}-4 x^{2}-27 x-18=0 ;[-4,3,1] \text { by }[-45,45,1]\end{aligned} $$

Step-by-Step Solution

Answer

The actual rational roots of the equation are -1, -3, and 1/2.

Step 1: Apply the Rational Zero Theorem

The Rational Zero Theorem states that if the coefficients of a polynomial are integers, then any rational zero must be of the form \( \pm\frac{p}{q} \) where \( p \) is a factor of the constant term (in this case, -18) and \( q \) is a factor of the leading coefficient (in this case, 2). By calculating these values, a list of possible rational roots can be created.

Step 2: List possible rational roots

Factors of -18 are: \( \pm1, \pm2, \pm3, \pm6, \pm9 \) and \(\pm18\). Factors of 2 are: \( \pm 1 \) and \( \pm 2 \). Testing combinations of \( p \) and \( q \), the possible rational roots are \( \pm1, \pm2, \pm3, \pm6, \pm9, \pm18, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{9}{2}, \) and \( \pm\frac{18}{2}.\)

Step 3: Graph the polynomial function

Next, graph the polynomial function \(2x^4+7x^3-4x^2-27x-18=0\) in the given viewing rectangle [-4,3,1] by [-45,45,1]. By observing the x-intercepts - the points where the graph crosses the x-axis - the actual rational roots of the equation can be determined.

Step 4: Identify the actual roots

Checking the graph, we can observe that the equation crosses the x-axis at -1, -3, and 1/2. Therefore, these three values are the actual rational roots of the equation.

Key Concepts

Polynomial Function

A polynomial function is an expression consisting of variables, exponents, and coefficients combined using addition, subtraction, and multiplication. It has the general form of \( a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0 \), where \( n \) is a non-negative integer, and \( a_n \) through \( a_0 \) are constants with \( a_n eq 0 \). The highest power of the variable determines the degree of the polynomial, and it is an essential aspect since it tells us how many solutions or roots the polynomial can have.

• The degree of the polynomial in this exercise is 4, which allows for up to four roots.
• Each polynomial feature, including terms, coefficients, and degree, plays a crucial role in graphing and solving.

In problems like the one provided, identifying these components is the first step in finding solutions, such as roots, through various methods, such as using the Rational Zero Theorem.

Rational Roots

Rational roots of a polynomial are roots that can be expressed as a fraction \( \frac{p}{q} \) where both \( p \) and \( q \) are integers, and \( q eq 0 \). The Rational Zero Theorem helps identify these roots by giving potential candidates that are rational numbers.

• The theorem requires listing factors of the constant term and the leading coefficient, then forming fractions from these factors.
• In the stated exercise, by applying the Rational Zero Theorem, we determine the potential rational roots: \( \pm1, \pm2, \pm3, \pm6, \pm9, \pm18, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{9}{2}, \pm9 \) and \( \pm\frac{18}{2}. \)

With potential roots in hand, testing or graphing helps verify which of these potential roots are actual solutions to the polynomial equation.

Graphing Polynomial Equations

Graphing polynomial equations provides a visual representation of the function, which helps in identifying the roots as the points where the graph intersects the x-axis.

• In this exercise, the polynomial is graphed within the viewing rectangle of \([-4,3,1]\) by \([-45,45,1]\), which essentially sets a specific window frame for the x and y-values on the coordinate plane.
• The graph of the polynomial function \(2x^4+7x^3-4x^2-27x-18=0\) shows where the polynomial crosses the x-axis.

Each intersection corresponds to a root, and by examining these points, we find the actual rational roots. The graph reveals the x-axis intercepts at \(-1, -3,\) and \(\frac{1}{2}\), confirming these values as the roots of the polynomial. This visualization lets us quickly verify potential roots from the Rational Zero Theorem by understanding the function's behavior.