Problem 22 - Precalculus Mathematics for Calculus

Question

Find the sum, difference, or product. $$5(3 t-4)-\left(t^{2}+2\right)-2 t(t-3)$$

Step-by-Step Solution

Answer

The expression simplifies to \(-3t^2 + 21t - 22\).

Step 1: Distribute the 5

Start by distributing the 5 across the terms inside the bracket in the expression \(5(3t - 4)\):\[5 \times 3t - 5 \times 4 = 15t - 20\].

Step 2: Distribute the -2t

Next, distribute \(-2t\) across the terms \((t - 3)\):\[-2t \times t - (-2t) \times 3 = -2t^2 + 6t\].

Step 3: Combine all parts

Combine all expressions from Steps 1 and 2 into the original equation and simplify it. This gives us: \[15t - 20 - (t^2 + 2) - 2t^2 + 6t\].

Step 4: Simplify Further

Combine like terms. Recognize that \(t^2\) terms, \(t\) terms, and constant terms should be grouped separately:- For \(t^2\): \(-1t^2 - 2t^2 = -3t^2\)- For \(t\): \(15t + 6t = 21t\)- For constants: \(-20 - 2 = -22\)Therefore, the simplified expression is: \(-3t^2 + 21t - 22\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept used in algebra that allows us to simplify expressions involving multiplication over addition or subtraction. Imagine you have an expression like \(a(b + c)\). The distributive property tells us that you can multiply each addend inside the parenthesis by \(a\), so it becomes \(ab + ac\). This is incredibly helpful for breaking down more complex expressions into simpler parts.

In our example, let's take \(5(3t - 4)\). We need to distribute the 5 to each term inside the parenthesis:

• First, multiply \(5\) by \(3t\) to get \(15t\).
• Next, multiply \(5\) by \(-4\) to get \(-20\).

Now, the expression \(5(3t - 4)\) becomes \(15t - 20\).
Understanding and using the distributive property efficiently will often be your first step when tackling complex polynomial expressions.

Simplifying Expressions

Simplifying expressions is all about making a mathematical expression more manageable by following some key algebraic rules. After using the distributive property, as in the earlier step, simplifying means cleaning up the expression further.

Initially, you may have expressions from different parts of the problem that you need to piece together. You aim to combine them into one expression, like we did:

• Start with the results from the distributive steps: \(15t - 20\) and \(-2t^2 + 6t\).
• You also have \(- (t^2 + 2)\). Apply distribution here as well: \(-t^2 - 2\).

Combine them all into a single expression:
\[15t - 20 - t^2 - 2 - 2t^2 + 6t\]
This expression might seem overwhelming at first glance, but by breaking it down and organizing it by similar types of terms, you make it easier to manage. The next critical step is combining like terms.

Combining Like Terms

Combining like terms is a technique that helps simplify expressions even further by grouping terms that are similar. "Like terms" are terms in a polynomial that have the same variables raised to the same powers. For example, \(3x\) and \(5x\) are like terms because they both have the variable \(x\) raised to the first power.

To combine like terms in our expression \(15t - 20 - t^2 - 2 - 2t^2 + 6t\), you can follow these steps:

• Group all the \(t^2\) terms: \(-t^2\) and \(-2t^2\). Adding these gives \(-3t^2\).
• Next, gather all \(t\) terms: \(15t\) and \(6t\). Combine these to get \(21t\).
• Finally, add the constant terms (the numbers without variables): \(-20\) and \(-2\), which equals \(-22\).

So, the simplified expression after combining like terms will be \(-3t^2 + 21t - 22\).
This process reduces the complexity of expressions, helping you to see the structure more clearly and enabling you to solve, graph, or further manipulate them efficiently.