Problem 132 - College Algebra Essentials

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I factored \(4 x^{2}-100\) completely and obtained \((2 x+10)(2 x-10)\)

Step-by-Step Solution

Answer

The statement makes sense because the factored form given by the student, albeit presented in a different order, correctly factors the given expression. However, the more standard presentation would be \((2x - 10)(2x + 10)\).

Step 1: Identifying a difference of squares

The given expression \(4x^2-100\) is a difference of squares. A difference of squares is an expression of the form \(a^2 - b^2\), which can be factored into \((a-b)(a+b)\). Here, \(a = 2x\) and \(b = 10\).

Step 2: Factoring the difference of squares

Factoring \(4x^2-100\) according to the property of difference of squares, we get \((2x - 10)(2x + 10)\). This is the correctly factored form of the expression.

Step 3: Comparison of the student's factoring with the correct factoring

The student's factored form is \((2x+10)(2x-10)\). Comparing this with the correct factored form \((2x - 10)(2x + 10)\), it is clear that the terms in the student's factored form are in the wrong order. However, the order doesn't affect multiplication, so the statement makes sense but needs to be presented as \((2x -10)(2x+10)\).

Key Concepts

College Algebra

College Algebra is a fundamental course that lays the groundwork for many advanced mathematics topics. It includes a variety of functions, graphs, equations, and inequalities. Students in college algebra explore how to manipulate algebraic expressions and solve equations that are critical for higher-level math and real-world problem solving.

The factoring of polynomials is a key skill in College Algebra, as it enables the simplification of equations and the ability to solve them. For example, the difference of squares is an important concept that comes up frequently, not only in College Algebra but also in calculus and other areas of mathematics and sciences. Mastering techniques such as factoring helps students move into more complex areas of study with confidence.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, which means the highest power of the variable, usually denoted as x, is 2. These equations can be written in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients and \( a \eq 0\).

Solvers of quadratic equations must be careful to account for all possible solutions. Techniques include factoring, completing the square, and using the quadratic formula. Factoring is often the quickest method, but it requires a keen eye to recognize patterns such as the difference of squares, which we see in the exercise \( 4x^2-100 \).

Polynomial Factoring

Polynomial factoring is the process of breaking down a polynomial into simpler components called factors, which when multiplied together, yield the original polynomial. There are various factoring techniques, including factoring by grouping, using the GCF (Greatest Common Factor), and recognizing special products such as the difference of squares.

In our exercise, we encounter a classic instance of factoring the difference of squares. The ability to factor polynomials effectively is crucial because it simplifies algebraic expressions and provides a more straightforward means of finding zeros or solving equations. Furthermore, polynomial factoring is used extensively in calculus, for instance, when integrating rational functions or finding limits.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can comprise numbers, variables, and operations. They do not contain equal signs, unlike equations. These expressions become even more interesting and complex when we introduce exponents and polynomials.

Factoring is one way to simplify these expressions and make them more manageable. In our example, \( 4x^2-100 \), we factored the expression into \( (2x-10)(2x+10) \), revealing a clearer structure of the original expression. Understanding how to manipulate and simplify algebraic expressions is vital for success in algebra and beyond, as it forms the foundation for more advanced problem-solving techniques used across various mathematical disciplines.