Problem 33 - Algebra 1: Concepts and Skills

Question

$$ (3 s-1)(s+2) $$

Step-by-Step Solution

Answer

The expanded form of \((3s-1)(s+2)\) is \(3s^2 + 5s - 2\).

Step 1: Distribute

First, distribute \(3s - 1\) across \(s + 2\). This means that every term in the first parentheses shall be multiplied by every term in the second parentheses. We get \(3s \cdot s + 3s \cdot 2 - 1 \cdot s - 1 \cdot 2\).

Step 2: Simplify

Now simplify the expressions resulting from the distributive property to obtain the expanded polynomial. This results in \(3s^2 + 6s - s - 2\).

Step 3: Combine Like Terms

Look for like terms and combine them. The terms \(6s\) and \(-s\) are like terms. So, their coefficients are combined, giving \(3s^2 + 5s - 2\).

Key Concepts

Understanding the Distributive Property

At the heart of polynomial expansion is the distributive property, a fundamental tool used to simplify mathematical expressions. It states that for any real numbers (or algebraic expressions) a, b, and c, the equation a(b + c) equals ab + ac holds true. To visualize this, imagine distributing something equally across various containers; similarly, you distribute a term across others in an algebraic expression.

Let's consider the problem \(3s-1)(s+2)\). Here, you apply the distributive property by multiplying each term in the first parentheses by each term in the second. Think of \(3s - 1\) as the 'something' to be distributed and \(s + 2\) as the 'containers'. The distribution process looks like this:

• Multiply \(3s\) by \(s\), and also by \(2\).
• Multiply \( -1\) by \(s\), and also by \(2\).

This approach helps unravel complex expressions to simpler forms, setting the stage for combining like terms and achieving a simplified expression.

Combining Like Terms

Once the distributive property has been applied, the next step usually involves combining like terms. But what are 'like terms'? They are terms within an algebraic expression that have identical variable parts regardless of their coefficients. For instance, in the expression \(3s^2 + 6s - s - 2\), the terms \(6s\) and \( -s\) are considered like terms because they both have the variable s raised to the first power.To combine them, you simply add or subtract their coefficients. In our example, adding \(6s\) to \( -s\) results in \(5s\), since \(6 - 1 = 5\). The expression then becomes \(3s^2 + 5s - 2\). This step is crucial for tidying up the expression and making it easier to understand and use in further calculations.

Simplifying Expressions

The final stage in polynomial expansion is simplifying the expression, which means writing the expression in its most reduced form. After distributing and combining like terms, you often end up with a polynomial that can still be simplified. Simplification helps to clear out the clutter in the expression, ensuring that it is presented in the most straightforward manner possible.

Simplifying \(3s^2 + 6s - s - 2\) leads to \(3s^2 + 5s - 2\) by combining like terms as discussed. There’s nothing more we can combine or reduce in this expression—it’s in its most simplified form. It's vital to simplify expressions because it makes subsequent calculations less error-prone and also makes it easier to interpret the results or graph the equations. Simplified expressions are a fundamental starting point for calculus, graphing functions, and solving equations.