Problem 18 - Elementary and Intermediate Algebra

Question

Factor. See Example 1 or Objective 1 $$x^{2}+7 x+10$$

Step-by-Step Solution

Answer

\((x + 2)(x + 5)\)

Step 1: Understand the Problem

The problem asks us to factor the quadratic expression \(x^2 + 7x + 10\). Factoring means writing it as a product of two binomials. We need to find two numbers that multiply to the constant term (10) and add to the linear coefficient (7).

Step 2: Find the Numbers

Look for two numbers that multiply to 10 (the constant term) and add up to 7 (the coefficient of the linear term). The pair that satisfies this is (2, 5) because \(2 \times 5 = 10\) and \(2 + 5 = 7\).

Step 3: Write the Binomial Factors

Using the numbers identified in Step 2, express the quadratic as a product of two binomials: \((x + 2)(x + 5)\).

Step 4: Verify the Factored Form

Expand \((x + 2)(x + 5)\) to ensure it equals the original expression: \((x + 2)(x + 5) = x^2 + 5x + 2x + 10 = x^2 + 7x + 10\). The factored expression is correct.

Key Concepts

Understanding Binomials in Factoring

In algebra, a binomial is an expression composed of two terms, typically connected by a plus or minus sign. Think of it as a mini version of a polynomial. When we are factoring a quadratic expression like \(x^2 + 7x + 10\), we aim to express this expression as a product of two binomials. Each binomial contributes either of its terms to form the quadratic expression when expanded.
For example, in our exercise, we ended with \((x + 2)(x + 5)\). Here, each binomial contributes to both the linear term (\(7x\)) and constant term (\(10\)) of the quadratic once you expand it.

• \(x + 2\) and \(x + 5\) are binomials.
• They combine through multiplication to return to the quadratic expression.

Recognizing and writing expressions as products of binomials is a key skill useful in simplifying complex problems. Understanding the relationships between terms is essential in mastering algebra.

Decoding Quadratic Expressions

Quadratic expressions are fundamental concepts in algebra, often taking the standard form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In our example, \(x^2 + 7x + 10\), we see that \(a = 1\), \(b = 7\), and \(c = 10\). Quadratics are called so because of the quadratic term \(x^2\), which is the highest power of \(x\) in the expression.
In this expression:

• \(7x\) is the linear term.
• \(10\) is the constant term.

The goal when factoring a quadratic expression is often to turn it into a product of simpler expressions, like binomials, to solve equations or to simplify calculations. Understanding the role of each term in the equation is crucial when attempting to factor it. Factoring quickly reveals the roots (solutions) of the quadratic equation when set to zero.

Polynomials and Their Parts

A polynomial is a broader term that describes expressions consisting of multiple terms made up of variables with whole-number exponents, coefficients, and constants. They can have one term (monomial), two terms (binomial), or more (like a trinomial or beyond). When working with quadratics such as our earlier example, \(x^2 + 7x + 10\), each component part – whether term, constant, or coefficient – plays a critical role in solving or transforming the polynomial.
In understanding polynomials:

• Recognize different types (monomial, binomial, trinomial)
• Identify each term by its coefficient and exponent

Polynomials can be added, subtracted, and multiplied to form new polynomials of various degrees. The process of factoring is just one method of simplifying these expressions, which makes them more manageable or reveals hidden properties such as roots. Ensuring you grasp the basic structure and form of polynomials is fundamental to excelling in algebra.