Problem 103 - Introductory Algebra for College Students

Question

Use the \([\mathrm{GRAPH}]\) or \([\mathrm{TABLE}]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If the graphs of \(y_{1}\) and \(y_{2}\) coincide, or if their corresponding table values are equal, this means that the polynomial on the left side has been correctly factored. If not, factor the trinomial correctly and then use your graphing utility to verify the factorization. \(x^{2}-5 x+6=(x-2)(x-3)\)

Step-by-Step Solution

Answer

In this case, since the polynomial \(x^{2}-5x+6\) was correctly factored as \((x-2)(x-3)\), the two functions \((y_{1}\) and \(y_{2})\) coincide both when graphed and when placed in a table.

Step 1: Plotting the Functions

Use your graphing utility to plot two separate functions: \(y_{1}=x^{2}-5 x+6\) and \(y_{2}=(x-2)(x-3)\).

Step 2: Comparing the Results

Now it's time to compare the graphs or table values of the function \(y_{1}\) and \(y_{2}\). If they coincide, i.e., the plotted functions overlap or their corresponding table values are equal, that would mean that \(x^{2}-5 x+6\) has been correctly factored as \((x-2)(x-3)\).

Step 3: Correct Factoring (If Needed)

If the graphs or table values of the two functions do not coincide, it means that the function has not been correctly factored. In such a case, find the right factors of the polynomial \(x^{2}-5 x+6\) and repeat the steps above to verify the correct factorization using your graphing utility. For the given expression, it can be factored as \((x-2)(x-3)\). Therefore, no further steps are required here.

Key Concepts

Understanding Graphing Utilities

Graphing utilities, such as graphing calculators or software programs like Desmos, GeoGebra, and others, are vital tools in visualizing mathematical concepts, particularly polynomial functions. They allow students to input equations and view the plotted graphs to analyze curve shapes, intersections, and roots. This visualization helps in understanding the behavior of polynomials and serves as a corroborative tool to confirm algebraic factorization.

When you use a graphing utility for factoring polynomials, you're essentially comparing two forms of the same function: its expanded form (like \(y_1 = x^2 - 5x + 6\)) and its factored form (like \(y_2 = (x-2)(x-3)\)). If both forms are equivalent, their graphs will superimpose on each other entirely, offering a visual confirmation that the factoring is correct. Similarly, checking the tables feature gives a numerical verification that for every input (or 'x' value), the output (or 'y' values) of both functions is identical.

Navigating Polynomial Equations

Polynomial equations represent the backbone of algebra and appear frequently across all levels of mathematics. They can describe a wide range of phenomena, from simple motions in physics to complex economic models. The standard form of a polynomial equation in single variable 'x' is written as \(an*x^n + a_{n-1)*x^{n-1}} + ... + a2*x^2 + a1*x + a0 = 0\), where 'an' to 'a0' are constants, and 'n' is a non-negative integer representing the degree of the polynomial.

Factoring polynomials is a process where the polynomial is expressed as a product of its factors. These could be binomials, trinomials, or other polynomials. The process often requires the use of different methods like grouping, the quadratic formula, synthetic division, or trial and error. Knowing how to factor polynomials enables you to solve for the roots (or zeroes) of the equation—the values of 'x' that make the polynomial equal to zero. These roots correspond to the points where the graph of the polynomial will intersect the 'x' axis.

How to Verify Factorization

Verifying factorization is an important step in ensuring that a polynomial has been broken down into its correct, simplest pieces. After factoring a polynomial, it's essential to check that the factored form is equivalent to the original polynomial. This can be done algebraically by multiplying the factors to see if the original polynomial is obtained or using a graphing utility.

By inputting both the original polynomial equation and the factored form into the graphing utility, and then observing the resulting graphs or table values, you can quickly determine if the factorization is correct. A mismatch in either the graphs or the table values indicates an error in the factoring process. This method provides a visual and numerical way to confirm the accuracy of your algebraic factorization, thus enhancing your understanding and proficiency in dealing with polynomial equations.