Problem 61 - Algebra 1

Question

Apply the distributive property. $$(q+4)(-3 q)$$

Step-by-Step Solution

Answer

The solution to \( (q+4)(-3q) \) is \( -3q^2 - 12q \)

Step 1: Apply the Distributive Property

Applying the distributive property to \( (q+4)(-3q) \) gives us \( -3q * q + -3q * 4 \). The distributive property is applied across the terms within the brackets onto the \( -3q \)

Step 2: Perform Multiplication

Next, we perform the multiplication operations in this equation. This gives us \( -3q^2 - 12q \).

Step 3: Simplify Result

The result from Step 2 is already in its simplest form.

Key Concepts

Multiplication of Polynomials

When you encounter polynomial expressions like \((q+4)(-3q)\), multiplying them can seem intimidating. However, the distributive property is your best friend here. It helps break down the problem into manageable pieces. So, how do we multiply polynomials?

First, you want to apply the distributive property, which involves multiplying each term in the first polynomial by each term in the second polynomial. For our exercise:\(-3q\) is multiplied by every term inside the parentheses \((q+4)\). This means:

• \(-3q \cdot q = -3q^2\)
• \(-3q \cdot 4 = -12q\)

Now, you're left with \(-3q^2 - 12q\) after performing these multiplications. It's crucial to follow these steps, as proper polynomial multiplication sets up the correct base for solving larger algebraic problems.

Simplifying Expressions

Once you've multiplied the polynomials, the next step is simplifying the expression, i.e., reducing it to its simplest form. Our example gives us \(-3q^2 - 12q\). This expression is already simplified because the like terms are combined effectively.

But, what does it mean to simplify? It means you must ensure that the expression has no further combining, factoring, or reducing needed. There are no unnecessary complications or longer forms.
Here are some common tips for simplifying:

• Combine like terms, where terms contain the same variable to the same power.
• Restrict the expression to non-negative exponents.
• Free the expression from any obvious common factors.

These are the rules. Make sure there are no further actions you can perform before considering it fully simplified. This clarity makes it easier to interpret the results of any polynomial operation.

Algebraic Expressions

Algebraic expressions such as \((q+4)(-3q)\) involve variables, numbers, and arithmetic operators like addition and multiplication. Understanding how these expressions work is fundamental in algebra. They usually represent relationships, patterns, or quantities.

Breaking down algebraic expressions helps in solving them. Here’s what you generally do:

• Identify the given variables and constants in the problem.
• Apply arithmetic operations, as per instructions (like the distributive property).
• Follow algebraic rules — such as operation hierarchy — to manage the terms correctly.

This often involves using properties and rules efficiently, like in our example through the distributive property. By getting comfortable with forms of algebraic expressions, you can solve equations, inequalities, and perform numerous calculus operations. Algebraic thinking is a vital skill in mathematics, and practicing these expressions builds and strengthens that ability.