Problem 26 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ (2 x-5)(7 x+2) $$

Step-by-Step Solution

Answer

The product of (2x - 5)(7x + 2) is \( 14x^{2} - 31x - 10 \).

Step 1: Distribution

First, distribute each term in the first binomial to each term in the second. \[ (2x \cdot 7x) + (2x \cdot 2) + (-5 \cdot 7x) + (-5 \cdot 2) \].

Step 2: Simplify each term

Notice that some of these terms are just multiplication of numbers, this can be simplified: \[ 14x^{2} + 4x - 35x - 10 \].

Step 3: Combine like terms

There are two 'x' terms here that can be combined.\[ 14x^{2} - 31x - 10 \].

Key Concepts

Binomials

A binomial is a type of polynomial, which is a mathematical expression consisting of variables and coefficients. A binomial itself is specifically made up of exactly two terms. These terms are typically separated by either a plus or minus sign. An example of a binomial is the expression \(2x - 5\). Here, there are two terms: \(2x\) and \(-5\). Another example is \(7x + 2\), which includes the terms \(7x\) and \(+2\).

Binomials are commonly used in algebraic expressions and equations. They play a significant role in polynomial multiplication. Understanding binomials is crucial, as it lays the groundwork for grasping more complex concepts in algebra, such as factoring and expanding polynomials.

Distribution

Distribution, also known as the distributive property, is a key concept in algebra. It allows you to multiply each term within a parenthesis by another term outside the parenthesis. This is essential when multiplying binomials.

For example, when distributing \((2x-5)\) and \((7x+2)\), you take each term in the first binomial and multiply it by each term in the second binomial. This involves the following steps:

• First, multiply \(2x\) by \(7x\), resulting in \(14x^2\).
• Then, multiply \(2x\) by \(2\) to get \(4x\).
• Next, \(-5\) is multiplied by \(7x\), resulting in \(-35x\).
• Finally, multiply \(-5\) by \(2\) to get \(-10\).

Distribution helps you to simplify expressions and is a vital tool in solving complex equations.

Like Terms

Like terms in an algebraic expression are terms that have the same variable raised to the same power. These terms can be combined to simplify an algebraic expression.

For example, in the expression \(14x^2 + 4x - 35x - 10\), the terms \(4x\) and \(-35x\) are like terms because they both contain the variable \(x\) raised to the first power. When you combine these like terms, you simply add or subtract their coefficients, which in this case results in \(-31x\). After combining, the simplified expression becomes \(14x^2 - 31x - 10\).

Recognizing and combining like terms is crucial for simplifying expressions and solving equations efficiently.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols such as addition and subtraction. An expression might look something like \(14x^2 - 31x - 10\).

These expressions do not have an equality sign like an equation does, which means they don't show a complete relationship and are not solved but rather simplified or evaluated. Algebraic expressions can be as simple as a single constant or variable, or more complex, like polynomials with multiple terms.

• They are used to generalize mathematical problems and represent real-world situations.
• Algebraic expressions can be manipulated using properties such as distribution and the combination of like terms.

Understanding how to work with these expressions is key to mastering higher-level algebra, as they form the basis of equations and functions.