Problem 90 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(77-96,\) simplify each algebraic expression. $$-3(4 x-5)$$

Step-by-Step Solution

Answer

-12x + 15

Step 1: Identify the Expression

In this case, the algebraic expression given is \(-3(4x - 5)\). This is an example of multiplication of a monomial \(-3\) by a binomial \(4x - 5\). The task is to simplify this expression using the distributive law.

Step 2: Apply the Distributive Property

Apply the distributive property, also known as the distributive law, which says that the multiplication of the monomial \(-3\) should be distributed across each of the terms inside the parentheses. So, it becomes: \(-3 * 4x - (-3 * 5)\).

Step 3: Simplify the Expression

Now simplify the expression by performing the multiplication: \(-12x - (-15)\). This simplifies further to \(-12x + 15\).

Key Concepts

Understanding the Distributive Property

Grasping the distributive property is essential when working with algebraic expressions. This rule is a cornerstone of algebra and allows us to simplify expressions by expanding multiplication over addition or subtraction. In essence, it tells us that when you multiply a number (or variable) by a sum or difference, you can 'distribute' the multiplication to each term within the parentheses. For example, if we have an expression like \( -3(4x-5) \), using the distributive property, we individually multiply \( -3 \) by \( 4x \) and \( -5 \) which results in \( -12x \) and \( +15 \) respectively.

Visualizing this can be helpful. Imagine that \( -3 \) is a factor that is being shared by both \( 4x \) and \( -5 \). It's similar to sharing a batch of cookies equally among friends; every term inside the parentheses gets an equal share of \( -3 \)

• Multiply \( -3 \) by \( 4x \) to get \( -12x \)
• Multiply \( -3 \) by \( -5 \) to get \(+15\)

Once these multiplications are performed, you combine the products to simplify the expression.

Monomial Multiplication Basics

When we talk about multiplying monomials, we are referring to taking a single-term algebraic expression (a monomial) and multiplying it by another term or polynomial. In our example, the monomial is \( -3 \) and it’s being multiplied by each term in the binomial \( 4x - 5 \). This process is straightforward: you multiply the coefficient (the numerical part) of the monomial with the coefficients of the other terms and apply the same operation to their variables.

Practical Steps for Monomial Multiplication:

• Identify the coefficients and variables.
• Multiply the coefficients together.
• Apply the arithmetic operations to the variables, adhering to exponent rules if applicable.

So when we multiply \( -3 \) and \( 4x \) we get \( -12x \) because \( -3 \times 4 = -12 \) and the variable \( x \) remains unchanged. It's much like combining like ingredients in a recipe, ensuring each is added in the correct quantity to produce the desired dish.

Binomial Simplification Strategies

Simplifying binomials involves reducing expressions with two terms into their simplest form. When a binomial is multiplied by a monomial, you perform the operations term-by-term, as shown in the distributive property. But simplification doesn't stop with distribution.

After applying the distributive property, look for like terms, which are the terms with the same variables raised to the same power. In the example \( -12x + 15 \), there are no like terms to combine, so this is already its simplest form. However, if we had an expression like \( -12x + 5x \), we could combine the \( x \) terms to further simplify it to \( -7x \).

Remember, binomial simplification is all about making an expression as easy to read and work with as possible, often to prepare it for further operations or to solve equations.

• If no like terms exist, ensure the expression is fully expanded and clear of parentheses.
• When like terms are present, combine them to simplify the expression.
• Keep the simplified expression in the same order: descending powers of variables if applicable.

This structured approach aids in clarity, ensuring each step moves towards a more digestible version of the original algebraic expression.