Problem 19 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ (x+7)(x+3) $$

Step-by-Step Solution

Answer

The product of \((x+7)\) and \((x+3)\) is \(x^2 + 10x + 21\).

Step 1: Apply the FOIL Method

This method stands for First, Outer, Inner, and Last. It is a way to multiply binomials. So first, multiple the first terms in each binomial \((x \times x)\), then the outer terms \((x \times 3)\), then the inner terms \((7 \times x)\), and finally, the last terms in each binomial \((7 \times 3)\). This will give four terms.

Step 2: Simplify the Resulting Terms

You would get: \(x^2\), \(3x\), \(7x\), \(21\). Combine like terms in order to simplify the expression. \(3x\) and \(7x\) are like terms because they are both multiples of \(x\). Add those together.

Step 3: Write Down the Final Answer

After combining like terms, the expression will simplify to \(x^2 + 10x + 21\). This is the final answer.

Key Concepts

Understanding Binomials

A binomial is a type of polynomial that consists of two terms connected by a plus or minus sign. In the given exercise, we have a binomial

• \((x + 7)\)

multiplied by another binomial

• \((x + 3)\).

Binomials are often part of algebraic expressions and are used in various mathematical calculations. These expressions need to be simplified or expanded to track changes when multiplied. Understanding binomials is vital because they form the foundation of more complex expressions in algebra. Recognizing that each part of a binomial contributes to expanding is the first step in evaluating expressions like these.

Expanding the World of Polynomials

Polynomials are algebraic expressions that consist of terms in the form of

• \(ax^n\) where \(a\) is a coefficient, and \(n\) is a non-negative integer.

In our exercise, after using the FOIL method, you get a polynomial expression

• \(x^2 + 3x + 7x + 21\).

The key to handling polynomials is understanding how to operate with these terms and combine them properly when possible. When multiplying binomials, the result is usually a polynomial, which often needs further simplification to reach a concise answer. The structure of polynomials makes them versatile tools to represent a wide range of mathematical problems, from simple everyday computations to complex engineering problems.

Simplifying Expressions Fully

Simplifying expressions is the process of making them easier to work with, usually involving fewer terms or more manageable numbers. In our case, after applying the FOIL method, simplifying involves combining the like terms

• \(3x\) and \(7x\)

to form a single term

• \(10x\).

This makes the expression more straightforward:

• \(x^2 + 10x + 21\).

This process doesn't change the value of the expression but makes it easier to understand and solve further tasks related to it. Simplifying expressions is a crucial skill in algebra as it allows problems to be expressed in more manageable forms, aiding in both understanding and computation.

Combining Like Terms

To create a simplified polynomial, it's essential to know how to combine like terms. Like terms are terms in an algebraic expression that have identical variables raised to the same power. In this exercise,

• \(3x\) and \(7x\)

are like terms because both contain the variable \(x\) to the power of 1. When combining them, you just add their coefficients:

• \(3 + 7\).

This is summed up to form the simplified term

• \(10x\).

Recognizing and combining like terms translates chaotic collections of terms into easy-to-handle expressions, simplifying calculations and problem-solving in algebra.