Problem 104 - Introductory Algebra for College Students

Question

My graphing calculator showed the same graphs for \(y_{1}=4 x^{2}-20 x+24 \quad\) and \(\quad y_{2}=4\left(x^{2}-5 x+6\right), \quad\) so I can conclude that the complete factorization of \(4 x^{2}-20 x+24\) is \(4\left(x^{2}-5 x+6\right)\).

Step-by-Step Solution

Answer

Using equivalence and basic mathematical operations, it can be confirmed that the two given expressions \(4 x^2 -20 x +24\) and \(4(x^2 -5x +6)\) are equivalent. Hence, it can be concluded that \(4 x^2 -20 x +24\) is completely factorized as \(4(x^2 -5x +6)\)

Step 1: Equate the two expressions

First, equate the two given expressions, i.e. \(4x^2-20x+24 = 4(x^2-5x+6)\). This sets up the problem for us to validate.

Step 2: Expansion of the right hand side

Now, expand the right side of the equation to check if it simplifies to the left side equation. The expansion becomes \(4*x^2 - 4*5x + 4*6 = 4x^2 - 20x + 24\).

Step 3: Compare the Simplified Expressions

Compare the simplified right hand side with the left hand side. This comparison shows that both sides of the equation match exactly, confirming the factorization is correct.

Key Concepts

Graphing Calculators in Algebra

Graphing calculators have become an indispensable tool in algebra, particularly when dealing with quadratic equations. These devices allow students and educators to quickly visualize the relationship between the variables of an equation, which can lead to a better understanding of the concept being taught.

A graphing calculator can display the graph of a quadratic equation like the one in our example, where the student compared the graphs of two algebraic expressions to conclude they are the same. This visual confirmation is a powerful way to comprehend that two forms of an equation are equivalent. In addition to graphing, calculators can also perform a variety of algebraic operations, such as factoring polynomials or solving systems of equations, making them incredibly versatile for students learning algebra.

Expanding Polynomials

Expanding polynomials is a key concept in algebra that involves rewriting expressions into a simplified or more standard form. For instance, when we take a factored polynomial, like the one in our provided solution, and multiply out the factors, we are expanding the polynomial.

Understanding how to expand polynomials correctly is crucial for verifying algebraic expressions, solving equations, and simplifying complex algebraic statements. In the example, the student expanded the polynomial by distributing the coefficient 4 across each term in the parentheses. The result of this expansion, as shown in the provided step-by-step solution, confirms that the original and factorized expressions are identical.

Verifying Algebraic Expressions

Verifying algebraic expressions is the process of confirming whether two expressions are equivalent. This task is fundamental in algebra to ensure that any factorization, simplification, or algebraic manipulation done is accurate.

In practice, this can be achieved by expanding polynomials, simplifying terms, and using logical reasoning to compare the results, just like in the exercise we reviewed. The step-by-step solution shows how, through careful expansion, the student is able to verify that the given expressions are indeed equal to each other. This reinforces the understanding of equivalence and identity in algebraic equations. Mastering verification methods is essential for students to confidently tackle and solve algebra problems.