Problem 23 - Algebra 2

Question

Factor each expression. $$ x^{2}+10 x-75 $$

Step-by-Step Solution

Answer

The factored form of the expression \(x^{2}+10x-75\) is \((x + 15)(x - 5)\).

Step 1: Determine the Structure

Identify the expression as a quadratic trinomial, which has the general form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In this case, \(a = 1\), \(b = 10\), and \(c = -75\).

Step 2: Find Two Numbers

Look for two numbers that multiply to \(ac = (1)(-75) = -75\) and that add to \(b = 10\). The numbers that fit this requirement are 15 and -5 because \(15 \times -5 = -75\) and \(15 + (-5) = 10\).

Step 3: Write the Pair as Binomials

Express the quadratic trinomial as a product of two binomials. Substitute the found pair of numbers into the binomials such that the expression takes the form of \((x + m)(x + n)\). For our expression, the factored form is \((x + 15)(x - 5)\).

Key Concepts

Quadratic Trinomial

When we speak of a quadratic trinomial, we’re diving into a basic yet significant part of algebra. This term refers to a polynomial with three terms that is also a quadratic, meaning its highest degree is two. The standard form for this type of expression is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\).

For instance, take \(x^2+10x-75\). Breaking this down, \(a\) equals 1 (since it's the coefficient of \(x^2\)), \(b\) is 10, and \(c\) is -75. Understanding this structure is pivotal as it sets the stage for factoring the expression.

Factoring a quadratic trinomial typically involves finding two binomials that, when multiplied, give you back the original expression. This can sometimes feel like a puzzle: you're looking for two numbers that fit exactly into the spaces left by \(a\), \(b\), and \(c\). It's like having a lock for which only the right combination will open it up. Once you've found this combination, you've effectively factored the quadratic trinomial. This foundational concept is a cornerstone of algebra and is extensively applied in various mathematical scenarios.

Factoring Expressions

Factoring expressions is a process similar to breaking a composite number down into its prime factors. In algebra, it’s about taking a complex expression and simplifying it into a product of simpler expressions.

Why would we want to factor expressions? For one, it can significantly aid in solving equations by transforming them into a more manageable form. Moreover, factoring is used in calculus for integration, finding limits, and so much more.

The method we use to factor an expression depends on its form. With quadratic trinomials, like the one in our exercise, we aim to find two binomials that, when multiplied, give us the original quadratic. This skill requires a blend of technique and intuition, where we combine our understanding of algebraic structures and number properties.

Binomial Products

Moving on to the concept of binomial products, these are the result of multiplying two binomials together. When dealing with factoring, our goal is to reverse this multiplication process—a task that can sometimes seem complicated.

A binomial is an algebraic expression containing two terms, such as \(x + m\) or \(x - n\). Their product forms a quadratic expression. The binomial theorem describes how to expand binomials raised to a power, but in our case, multiplication of two simple binomials involves using the distributive property or the FOIL (First, Outer, Inner, Last) method.

To factor a quadratic trinomial, we look for a pair of binomials whose product is the original expression. It’s an essential skill because it leads to the understanding of how to solve more complex equations and how different algebraic expressions are interconnected.

Algebraic Structures

Last but not least, let's discuss the importance of understanding algebraic structures as it relates to factoring. Algebraic structures include sets and operations defined on these sets that follow specific rules. These structures provide the framework within which we operate when we manipulate algebraic expressions.

In the realm of quadratic trinomials, recognizing the structure is the key first step towards successful factoring. It tells us what kind of solutions we should expect or even if factoring is possible. The rules and properties of algebraic structures guide us in manipulating expressions, allowing us to convert complex problems into simpler forms that are easier to analyze and solve.

Understanding these structures underpins much of mathematics and enables students to grasp more advanced concepts. It's like learning the grammar of a language, providing the ability to form correct and meaningful sentences—or in this case, to solve mathematical problems accurately and efficiently.