Problem 142 - 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 not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$4 x^{2}-12 x+9=(4 x-3)^{2} ;[-5,5,1] \text { by }[0,20,1]$$

Step-by-Step Solution

Answer

The correct factorization of the polynomial \(4x^{2} - 12x + 9\) is \(4(x - \frac{3}{2})^{2}\).

Step 1: Comparing the Expanded and Factored Forms

The left side of the equation is \(4x^{2} - 12x + 9\). The right side is \((4x - 3)^{2}\). Expanding the right side, \((4x - 3)^{2}\) becomes \(4^2x^2 - 2*4*3x + 3^2 = 16x^2 - 24x + 9\). The left and right side are not the same, thus the factorization is not correct.

Step 2: Correctly Factoring the Polynomial

The polynomial \(4x^{2} - 12x + 9\) can be factored as follows: \(4(x^{2} - 3x + \frac{9}{4})\). This can be further factored into: \(4(x - \frac{3}{2})^{2}\), which is the correct factorization of the polynomial.

Step 3: Graphing and Verifying the Factorization

Upon graphing both the original function \(4x^{2} - 12x + 9\) and the correctly factored function \(4(x - \frac{3}{2})^{2}\), you should observe that their graphs are identical. This confirms that the factorization is indeed correct.

Key Concepts

Graphing Utilities in Polynomial Equations

Graphing utilities are a powerful tool that can help visualize polynomial equations. These utilities aid in identifying factors and verifying solutions. By using features such as the "GRAPH" or "TABLE" functions, you can easily plot the graph of a polynomial equation.
This visualization allows you to check if a polynomial has been factored correctly by comparing the graphs of the original and factored forms. For instance, in our original equation, if both sides were correctly factorized, their graphs should be the same. If not, it prompts us to re-evaluate our factorization.
Using graphing utilities can save time and reduce errors, making them an invaluable resource for tackling polynomial equations.

Understanding Polynomial Equations

Polynomial equations are expressions consisting of variables and coefficients, which involve operations of addition, subtraction, multiplication, and non-negative integer exponents. The example in our exercise is a second-degree polynomial: \(4x^{2} - 12x + 9\).
These equations can often be solved, simplified, or re-arranged through factorization. The solution of a polynomial equation is finding the values of the variable that satisfy the equation. In practice, this often involves transforming the equation into a product of simpler expressions.
Recognizing the degree and the leading coefficient can help predict the general shape of a polynomial's graph, which is essential when using graphing utilities for analysis.

The Importance of Correct Factorization

Correct factorization transforms a complex polynomial into simpler, more manageable pieces, making it easier to analyze or solve. In the original exercise, wrongly identifying the factorization led to inconsistent results between the expanded and factored forms.
Correct factorization involves correctly restructuring a polynomial into its simpler components, as demonstrated with \(4x^{2} - 12x + 9\) factored into \(4(x - \frac{3}{2})^{2}\). This involves using techniques such as grouping and applying formulas like the square of a binomial.
Always verify your factorization by expanding it back to its original form or using graphing utilities to ensure it aligns with the polynomial's graph. This not only confirms accuracy but also reinforces understanding of the factorization process.