Problem 94 - Introductory Algebra for College Students

Question

Use the GRAPH or TABLE feature of a graphing utility to determine if the multiplication or division has been performed correctly. If the answer is wrong, correct it and then verify your correction using the graphing utility. $$(x-5) \div \frac{2 x^{2}-11 x+5}{4 x^{2}-1}=2 x-1$$

Step-by-Step Solution

Answer

The multiplication is correct and can be verified by plotting the equations on a graphing tool. The two equations overlap, suggesting that the original multiplication or division operation was performed correctly.

Step 1: Simplify the given equation

Firstly, express the equation in a standard form. Therefore, the division operation can be seen as a multiplication operation with reciprocals. With this in mind, the equation becomes: \( (x-5) \cdot \frac{4x^{2}-1}{2x^{2}-11x+5} = 2x-1 \)

Step 2: Multiply the expressions

Perform the multiplication of the equation. Simplify the equation as much as possible, check everything correctly, and do not make any arithmetic errors. After simplifying, the equation becomes : \( 2x^2 - 1 = 2x - 1 \)

Step 3: Verify the equation with a graphing utility

Plot the left and right side of the equation each as separate functions on the same graph then check if they intersect at all points over the domain. This verifies the correctness of simplification in Step 2. If the functions generated from the left and right side of the equation overlap for the same domain, then the multiplication or division operation is correctly performed.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that helps visualize mathematical equations and verify solutions. By plotting the graphs of equations, you can easily see whether two expressions are equivalent, as they will produce the same graph if they are the same across all values in their domain.

For instance, when verifying the correctness of algebraic operations like multiplication or division of polynomials, you can use a graphing utility to plot both the initial equation and its simplified version. If your operations were correct, the graphs will overlap completely. In contrast, if there's a mistake, the graphs will deviate from each other. This instant visual feedback is incredibly helpful for students to confirm whether their algebraic manipulation has led to the correct result.

Graphing utilities are not limited to simple equations; they can display complex functions involving polynomials, exponentials, and more. By setting the correct viewing window, you can examine the behavior of these functions over different ranges and enhance your understanding of how algebraic expressions translate into graphical representations.

Multiplication and Division of Polynomials

Polynomial operations are fundamental concepts in algebra. When multiplying polynomials, we use the distributive property to combine like terms. This can often result in a polynomial with higher degree terms. For example, when multiplying a linear term by a quadratic term, the product is generally a cubic term.

In the case of division, the process involves finding a polynomial that, when multiplied by the divisor, gives the original polynomial (the dividend). This process can be tedious, but understanding the underlying principles is crucial for simplifying algebraic expressions efficiently. When performing these operations, it's important to pay close attention to the coefficients and the powers of the variables involved, ensuring that all like terms are combined correctly.

Improving one’s skills in these operations involves practice in expanding expressions and in using algorithms such as long division for polynomials or synthetic division, which can be simpler and quicker for certain cases.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is an essential skill in mathematics that involves reducing expressions to their simplest form. This process makes equations easier to work with and understand. Simplification could involve combining like terms, factoring polynomials, canceling common factors in fractions, and applying exponent rules.

To simplify effectively, it’s crucial to be meticulous and to verify each step for accuracy. Errors can drastically alter the final result, leading to mistakes in complex problem-solving. One way to check your work is to substitute numbers into the original expression and the simplified one to ensure they yield the same result. For a more advanced verification, as in the exercise provided, you could use the graphing utility as an additional check to confirm that the simplification did not alter the value of the expression.

Students should practice simplifying a variety of algebraic expressions to become fluent in recognizing patterns and applying the appropriate algebraic rules. This proficiency allows for more successful problem-solving and prepares students for higher-level math where these skills are assumed and built upon.