Problem 144 - 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. $$\begin{aligned} &6 x^{2}+10 x-4=2(3 x-1)(x+2) ;[-5,5,1] \text { by }\\\ &[-20,20,2] \end{aligned}$$

Step-by-Step Solution

Answer

The correctness of the factorisation can be confirmed or denied by comparing the graphs of both the original equation, \(6x^{2}+10x-4\), and the factored equation, \(2(3x-1)(x+2)\). If the factorisation was incorrect, factor the equation correctly and graph it to verify correctness.

Step 1: Graph the Given Polynomial

First, graph the function \(6x^{2}+10x-4\) with the domain and range provided [-5,5,1] by [-20,20,2]. Pay close attention to how the curve behaves and its crossing points on the x-axis, these could be roots of the polynomial.

Step 2: Factor and Graph

Factor the given polynomial, which was provided as \(2(3x-1)(x+2)\). Graph this function as well, using the same domain and range as before.

Step 3: Compare the Graphs

Compare both graphs. If the two graphs correspond exactly, then the given polynomial was indeed factored correctly. If the graphs don't match, that's indicative a wrong factorisation.

Step 4: Correct Factorisation (if needed)

In case the factorisation was not correct, find the correct factorisation. In order to do this, you may need to use factorisation methods such as factoring by grouping, factoring out common factors, or using the quadratic formula. Once found, graph the factorised equation to confirm your solution.

Key Concepts

Understanding the Use of Graphing Utilities

Graphing utilities are digital tools—software or graphing calculators—that enable students and mathematicians to visualize complex algebraic equations. These tools create a graph based on the equation provided, helping users to comprehend how variables interact and the overall shape of the function.

When working with polynomials, graphing utilities become powerful as they visually display the roots where the graph crosses the x-axis. These are the values for which the polynomial equals zero, also known as the 'zeros' of the function.

• The first step is often to input the polynomial into the graphing utility to see its shape and intercepts.
• Then, the domain and range settings can be adjusted to zero in on the relevant sections of the graph.
• With these visuals, it becomes easier to hypothesize potential factors of the polynomial.

During this process, graphing utilities not only provide insights into the factorization but also validate algebraic work, making them indispensable in the study of polynomials.

Mastering the Art of Factoring Polynomials

Factoring polynomials is a foundational skill in algebra that requires breaking down complex expressions into products of simpler factors. The objective is to represent the polynomial in a way that reveals its roots directly.

• Start by looking for a Greatest Common Factor (GCF) that can be factored out.
• If the polynomial is quadratic, check if it can be factored into a product of two binomials.
• For more complex polynomials, techniques like factoring by grouping, synthetic division, or special product patterns might be required.

For example, in the equation provided, factoring the polynomial has led to the representation of \(2(3x-1)(x+2)\). This suggests that the roots could be at x-values solving \(3x-1=0\) and \(x+2=0\).

The factoring process is verified when these roots are seen to correspond with the x-intercepts of the graph of the polynomial. Through practice, identifying factorable polynomials and applying these methods becomes second nature.

Verifying Factorization Algebraically

Once a polynomial is factored, it's crucial to verify the factorization algebraically to ensure accuracy. This process involves several checks:

• Ensure that when the factored form is expanded, it simplifies to the original polynomial.
• Substitute the roots derived from the factored form into the original polynomial to check whether they result in a zero value, confirming them as true roots.
• Consider all possible factors and test them to see if they lead to the original polynomial.

Using the example from the exercise, you would expand \(2(3x-1)(x+2)\) and compare the result to the original polynomial \(6x^{2}+10x-4\) to confirm if the factorization is correct.

Verifying factorizations algebraically is a critical step that instills confidence in the outcomes derived from graphical interpretations, ensuring that the solutions are mathematically sound.