Problem 12 - Beginning and Intermediate Algebra

Question

Multiply. $$-12 m(11 m-4)$$

Step-by-Step Solution

Answer

The short answer is \(-12m(11m - 4) = -132m^2 + 48m\).

Step 1: Identify the monomial and binomial

In this exercise, we have a monomial, \(-12m\), and a binomial, \(11m - 4\).

Step 2: Distributive Property

Now, we will apply the distributive property by multiplying the monomial \(-12m\) by each term of the binomial \(11m - 4\).

Step 3: Multiply the first term of the binomial

First, we'll multiply the monomial \(-12m\) by the first term of the binomial, \(11m\): \[ (-12m) \times (11m). \] Now we multiply the coefficients, and multiply the variables: \((-12) \times (11) = -132\) and \(m \times m = m^2\). So, we get: \[ (-12m) \times (11m) = -132m^2. \]

Step 4: Multiply the second term of the binomial

Next, we'll multiply the monomial \(-12m\) by the second term of the binomial, \(-4\): \[ (-12m) \times (-4). \] Now we multiply the coefficients: \((-12) \times (-4) = 48\), and we keep the variable \(m\). So, we get: \[ (-12m) \times (-4) = 48m. \]

Step 5: Combine the results

Now that we've multiplied the monomial \(-12m\) by each term of the binomial \(11m - 4\), we can combine our results: \[ -12m(11m - 4) = -132m^2 + 48m. \] Thus, the final answer is \(-132m^2 + 48m\).

Key Concepts

Monomial

A **monomial** is a type of algebraic expression that consists of a single term. It's composed of numbers, variables, or both, multiplied together. In simple terms, it's like having one package, whether it's a number, a letter, or both in a multiplication dance. For example, in our problem, the monomial is \(-12m\).
If you break it down:

• \(-12\) is the coefficient, which is the numerical part of the monomial.
• \(m\) is the variable, which can represent various numbers.

Understanding monomials is crucial because they are the building blocks of more complex expressions, like binomials and polynomials. It's like knowing one Lego block to create a fantastic figure!

Binomial

A **binomial** is an algebraic expression that consists of two terms separated by a plus or minus sign. It's like having two different packages tied together. In our exercise, the binomial is \(11m - 4\).
Here's what it looks like:

• The first term is \(11m\) which consists of the coefficient \(11\) and the variable \(m\).
• The second term is \(-4\) which is a constant because it doesn’t have any variables.

Binomials are very common in algebra and often appear in operations such as addition, subtraction, and multiplication. Understanding binomials helps you tackle polynomial expressions and solve equations effectively. They are a step up from monomials and are essential in algebraic operations.

Distributive Property

The **Distributive Property** is a rule in algebra that allows us to multiply a single term by a sum or difference within a parenthesis. It's like the rule that tells us how to distribute chocolate evenly at a party! 🌟
In our problem, we have to multiply the monomial \(-12m\) by each term in the binomial \(11m - 4\). The distributive property lets us do that:
Simply distribute the monomial to both terms inside the binomial:

• First, multiply \(-12m\) with \(11m\).
• Next, multiply \(-12m\) with \(-4\).

This property is useful because it helps us break down complex expressions and simplify them step-by-step.

Algebraic Expressions

**Algebraic expressions** are mathematical phrases that include numbers, variables, and operations (such as addition and multiplication). They are like sentences but written in the language of math! Expressions can range from simple to complex, involving various parts, such as monomials and binomials.
Here's what makes up an algebraic expression:

• **Variables**: symbols like \(x\) or \(m\) that stand in for numbers.
• **Coefficients**: numbers that multiply variables, such as \(-12\) in \(-12m\).
• **Constants**: fixed numbers such as \(-4\).

In our example, we worked with a complete algebraic expression by multiplying the monomial by the binomial, resulting in a new algebraic expression \(-132m^2 + 48m\). Learning to work with these expressions is essential for mastering algebra and solving real-world problems!