Problem 17 - College Algebra and Calculus: An Applied Approach

Question

Find all the zeros of the function and write the polynomial as a product of linear factors. $$g(x)=x^{3}-6 x^{2}+13 x-10$$

Step-by-Step Solution

Answer

The zeros of the function are \(x = 1, 2, 5\). The polynomial can be expressed as a product of linear factors as \(g(x) = (x - 1)(x - 2)(x - 5)\).

Step 1: Setting the function equal to zero

This involves solving for \(x\) in the equation \(g(x) = 0\), i.e., \(x^{3} - 6x^{2} + 13x - 10 = 0\).

Step 2: Finding the roots of the equation

One can plot the graph or use numerical methods to approximate the roots. For exact answers, one may use the cubic formula for finding roots of a cubic polynomial, but for simplicity, one can see that 1, 2, and 5 are roots of this equation, either by trial and error or using synthetic division.

Step 3: Writing the polynomial as a product of linear factors

After identifying the roots, the function is expressed as \( g(x) = (x - 1)(x - 2)(x - 5) \). Each factor in the parenthesis represents a root of the equation \(g(x) = 0\).

Key Concepts

Solving Cubic Equations

The challenge of solving cubic equations, such as finding the roots of the function
g(x) = x^3 - 6x^2 + 13x - 10,
is a fundamental skill in algebra. A cubic equation is one where the highest exponent of the variable is three. To solve these equations, we set the function equal to zero and then look for values of x that satisfy the equation. One method is to identify possible rational roots using the Rational Root Theorem, which tells us to look at factors of the constant term (in this case, -10) over factors of the leading coefficient (which here is 1). Through trial and error or methods like synthetic division, we can test potential roots.
If a tested value is indeed a root, we can then divide the cubic polynomial by the linear factor associated with that root ((x - root)), simplifying the problem to solving a quadratic equation. While this process is systematic, it can be labor-intensive without a graphing tool or other computational aids.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into the product of its factors, much like factoring numbers into prime factors. A crucial application of polynomial factorization is in solving polynomial equations. For the function
g(x) = x^3 - 6x^2 + 13x - 10,
once we’ve determined its roots, we can write the original polynomial as the product of linear factors associated with these roots. Each factor is in the form of ((x - root)), where 'root' is a solution to the equation g(x) = 0. Factorization is advantageous since it allows us to visualize all the zeros of the function and provides a more straightforward way to graph the function. Additionally, understanding how to factor polynomials makes it easier to simplify complex algebraic expressions and solve rational equations.

Roots of Equations

The roots of an equation are the values of x that make the equation true when the equation is set to zero. In the context of the given cubic function,
g(x) = x^3 - 6x^2 + 13x - 10,
the roots are the x-values where the function intersects the x-axis on a graph. These are the solutions to the equation g(x) = 0. Understanding the concept of roots is critical because it describes the behavior of the function and aids in graph interpretation. Roots are not only crucial in algebra but also in calculus, where they assist in finding where functions increase, decrease or have inflection points. In other fields such as physics and engineering, roots can represent equilibrium points or conditions for system stability. Mathematical tools such as the Quadratic Formula or the Cubic Formula provide systematic ways to find roots, but for higher-degree polynomials, numerical methods or graph-based approaches may be necessary.