Problem 32 - Algebra and Trigonometry

Question

Find the sum, difference, or product. \(5(3 t-4)-2 t(t-3)\)

Step-by-Step Solution

Answer

The expression simplifies to \(-2t^2 + 21t - 20\).

Step 1: Distribute in the First Term

First, apply the distributive property to the expression inside the parentheses of the first term, which is \(5(3t - 4)\). This means multiplying \(5\) by each term inside the parentheses.\[5(3t - 4) = 5 \cdot 3t - 5 \cdot 4 = 15t - 20\]

Step 2: Distribute in the Second Term

Now, distribute the \(-2t\) across the terms inside the parentheses of the second term, \(-2t(t - 3)\). Multiply \(-2t\) by each term in the parentheses.\[-2t(t - 3) = -2t \cdot t + (-2t) \cdot (-3) = -2t^2 + 6t\]

Step 3: Combine Like Terms

Combine the results from Step 1 and Step 2. Add or subtract the coefficients of like terms (terms with the same variable raised to the same power).The expression from the two distributed terms is:\[15t - 20 - 2t^2 + 6t\]Combine the terms: - Combine \(t\)-terms: \(15t + 6t = 21t\)- The expression becomes:\[-2t^2 + 21t - 20\]

Step 4: Final Simplified Expression

The simplified expression after combining all like terms is:\[-2t^2 + 21t - 20\]This cannot be further simplified since there are no more like terms to combine or further operations required.

Key Concepts

Distributive Property

When dealing with expressions, the distributive property is very useful. It allows us to multiply a single term by each term inside a set of parentheses. This is particularly helpful when you have expressions like \(5(3t - 4)\). To apply the distributive property here, multiply the number outside the parentheses, in this case, 5, by each individual term inside the parentheses:

• Multiply 5 by \(3t\) to get \(15t\).
• Multiply 5 by \(-4\) to get \(-20\).

Therefore, \(5(3t - 4) = 15t - 20\). This provides us with a simplified expression that no longer contains any parentheses.If there are more parentheses in another section of the expression, like \(-2t(t - 3)\), you'd follow the same process. Here, you'd multiply \(-2t\) by each term inside its parentheses as well. Distributing helps transform expressions into a form that is easier to work with.

Combining Like Terms

After applying the distributive property, you often end up with an expression full of terms. Some of these terms can be combined to make the expression simpler. This process is called "combining like terms."Like terms are terms that have the exact same variable part. For instance, \(15t\) and \(6t\) are like terms because they both have the \(t\) variable. Similarly, constant terms (like \(-20\)) and multiple instances of squared terms (like \(-2t^2\)) are considered like terms if they share the same degree in variable form.When you have the expression \(15t - 20 - 2t^2 + 6t\):

• The \(t\) terms \(15t\) and \(6t\) can be combined to make \(21t\).
• The \(-20\) stays the same because it's a constant term by itself with no other like term to combine with.
• The \(-2t^2\) is also on its own, with no other \(t^2\) terms available.

Combining terms makes your expression tidier and easier to understand, resulting in \(-2t^2 + 21t - 20\).

Simplifying Expressions

Simplifying an expression is about making it as concise as possible without changing its value. After distributing and combining like terms, the last step is to check if there's any further simplification possible.In the expression \(-2t^2 + 21t - 20\), scrutiny tells us that there are no like terms remaining and no further operations needed. Therefore, this expression is in its simplest form. Here are some things to keep in mind:

• Double-check that all terms capable of being combined have indeed been merged.
• Look for any common factors across the entire expression that could simplify it further.
• Ensure signs are correctly assigned to each term during operations.

Simplifying expressions requires careful revisiting of the whole process to confirm that the smallest possible form is achieved, which is especially important for accurate homework completion or during examinations.