Question

\(g(x)=x^5-3 x^4+2 x-4\)

Short answer

To find the roots of the polynomial \(g(x)=x^5-3x^4+2x-4\), we use Rational Root Theorem and check each potential root by evaluating the function at that point. A root is valid if the value of the function at that point is zero.

Step-by-step solution

Understanding the Function

The function is a 5th degree polynomial and contains five terms, which could each have its own root. The first goal in understanding the function is to visualize it, and this could be done graphically.

Finding Potential Roots

Rational roots of the polynomial can be identified by applying the Rational Root Theorem. This could be done by taking all divisors of the constant term (in this case, -4), and considering all variations of positive and negative versions of these factors as potential roots. They could be +/- 1, 2, 4.

Checking Potential Roots

For each potential root, the value of the polynomial function at that root should be calculated. This can be done by replacing `x` in \(x^5-3x^4+2x-4\) with the potential root and calculating the result. If the result is 0, then that value of `x` is a root of the polynomial.

Confirming Roots

Finally, any root that leads to an output of zero for the function is a valid root. It is possible for the polynomial to have more roots, but in this case, only the real roots are considered.

Key concepts

Polynomial Functions

A polynomial function is an algebraic expression that consists of variables and coefficients, constructed using arithmetic operations like addition, subtraction, and multiplication. The general form of a polynomial can be \[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \]where \( a_n, a_{n-1}, \ldots, a_0 \) are constants with \( a_n eq 0 \) and \( n \) is a non-negative integer indicating the polynomial's degree.
In the exercise, we have the polynomial \( g(x) = x^5 - 3x^4 + 2x - 4 \), which is a 5th-degree polynomial. This means it can potentially have up to five roots.
Polynomial functions are significant because they can represent a broad range of real-world scenarios, including natural phenomena and business models. Understanding these functions involves analyzing their degree, terms, and behavior as the variable changes over real numbers.

Roots of Polynomials

The roots of a polynomial are the solutions to the equation \( P(x) = 0 \). They represent the values of \( x \) that make the entire polynomial equal to zero. Discovering these roots is key to understanding the polynomial's behavior.
The Rational Root Theorem provides a tool to find potential rational roots. According to this theorem, any rational root, expressed in the simplest form \( \frac{p}{q} \), must have \( p \) as a factor of the constant term \( a_0 \) and \( q \) as a factor of the leading coefficient \( a_n \).
In our exercise, the polynomial \( g(x) = x^5 - 3x^4 + 2x - 4 \) has a constant term of -4 and a leading coefficient of 1 (from \( x^5 \)). Thus, potential roots can be \( \pm 1, \pm 2, \pm 4 \). To determine if these are actual roots, you substitute these values into the polynomial and see if they yield a zero.

• If \( g(a) = 0 \), then \( a \) is a root.
• This process helps identify real roots, which are crucial for further analysis and graphing.

Graphing Polynomials

Graphing polynomial functions provides a visual representation of their behavior and helps to identify properties such as roots, intercepts, and turning points.
For the polynomial \( g(x) = x^5 - 3x^4 + 2x - 4 \), graphing allows you to observe where the polynomial crosses the x-axis, indicating the roots of the function. Since it's a 5th-degree polynomial, expect complex curves with several ups and downs.
Key aspects to consider when graphing:

• Roots and Intercepts: Each x-intercept corresponds to a real root of the polynomial. These are the \( x \)-values where the graph meets the x-axis, which were identified using methods like the Rational Root Theorem.
• End Behavior: For odd-degree polynomials like this, the ends of the graph will head in opposite directions. As \( x \) approaches infinity, the function's direction is determined by the leading term \( x^5 \), which contributes to predictable end behavior.
• Turning Points: A 5th-degree polynomial can have up to 4 turning points, places where the graph changes direction. These are also significant as they influence the overall shape of the graph.

Graphing helps to connect the analytical understanding of polynomials to their visual structure, enhancing comprehension and application in various contexts.