Problem 67 - Algebra 1

Question

COMBINING LIKE TERMS Apply the distributive property. Then simplify by combining like terms. $$ 3 t(t-5)+6 t^{2} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(9t^2 - 15t\).

Step 1: Apply Distributive Property

Apply the distributive property to \(3t(t-5)\) to remove parenthesis. To do so, multiply \(3t\) with each term inside the parenthesis individually. This gives \(3t * t - 3t * 5 = 3t^2 - 15t\)

Step 2: Combine Like Terms

Combine like terms of the result from Step 1 with the term \(6t^2\) from the original problem. All 't squared' terms (\(3t^2\) from step 1 and \(6t^2\)) are similar and can be combined into one term, which gives \(3t^2 + 6t^2 - 15t\)

Step 3: Simplify the Expression

Now, add the coefficients of the like terms to simplify the expression to a more reduced form. So, \(3t^2 + 6t^2 - 15t\) becomes \(9t^2 - 15t\).

Key Concepts

Distributive Property

Understanding the distributive property is fundamental when working with algebraic expressions. It's one of the key tools that enables us to simplify complex expressions.

Let's break it down in simple terms. Put plainly, the distributive property lets us multiply a single term by each term inside a parenthesis. For example, if we have the expression \(a(b + c)\), we would distribute the \(a\) across the \(b\) and \(c\), giving us \(ab + ac\).

When it comes to the exercise \(3t(t-5)\), we utilize the distributive property by multiplying \(3t\) with \(t\) to get \(3t^2\), and then with \( -5\) to get \( -15t\). The process transforms our expression to \(3t^2 - 15t\), paving the way for further simplification. It is like unpacking a box, where multiplication unwraps the terms so they can freely combine with other like terms.

Simplify Expressions

Once we've distributed the terms, the path is set to simplify the expression. This means merging like terms to make the algebraic statement as easy as possible. Like terms are the components in an algebraic expression that have the exact same variable parts raised to the same power. They can be friends and hang out together on the number line.

In our example, after applying the distributive property, we got \(3t^2 - 15t + 6t^2\). Notice we have \(3t^2\) and \(6t^2\); these are like terms. We can combine them by adding their coefficients, which are the numbers in front of the \(t^2\). This simplifies the expression to \(9t^2 - 15t\).

By doing this, we’re slimming down the expression, reducing any excess algebraic 'weight' – that's one fitness goal we can make happen mathematically!

Algebraic Expressions

When we talk about algebraic expressions, imagine a story told through numbers, variables, and operations. It's not a fixed narrative; the characters (variables) can change, and the plot (the expression) can evolve with different inputs.

Our initial expression \(3t(t-5) + 6t^2\) is an example of an algebraic expression. It consists of terms, which are the building blocks. Just like words form a sentence, in algebra, these terms come together to form expressions.

The goal is to write this story as concisely as possible. That's why we combine like terms and simplify. We're effectively editing the story to make it easier to read and understand. By the end of our problem, after distributing and combining like terms, we've refined our expression to \(9t^2 - 15t\), a much cleaner and more efficient rendition of the mathematical tale we started with.