Problem 20 - Algebra 2 and Trigonometry

Question

Perform the indicated operations and write the result in simplest form. \(b^{2}+b(3 b+5)\)

Step-by-Step Solution

Answer

The simplest form is \( 4b^2 + 5b \).

Step 1: Distribute within the Parenthesis

We start with the expression: \[ b^2 + b(3b + 5) \]In this step, distribute the single term \( b \) across the terms inside the parenthesis:\[ b^2 + 3b^2 + 5b \]

Step 2: Combine Like Terms

Now, we combine like terms. Add the \( b^2 \) terms together and write \( 5b \) at the end:\[ (b^2 + 3b^2) + 5b = 4b^2 + 5b \]

Step 3: Write the Final Simplified Expression

The expression \( 4b^2 + 5b \) is already in its simplest form because it has no like terms to combine further or any common factors in all terms to factor out.

Key Concepts

Simplifying Expressions

Simplifying expressions is like organizing your room – you want everything in its right place with no clutter. When you simplify, you're aiming to make an expression clear and concise. This ensures there are no unnecessary or repetitive parts. A simplified expression is easier to use in equations and calculations.

• Begin by doing any operations inside parentheses first.
• Next, deal with exponents or powers, if present.
• Lastly, perform multiplication or division before moving on to addition and subtraction.

In our example, we simplify by first distributing the terms, then combining like terms. Remember, simplifying can make solving more complex problems easier down the track.

Distributive Property

The distributive property is a handy tool to help break down complex expressions into manageable parts. Think of it as sharing or distributing a number or variable equally across a sum or difference in parentheses.
When you apply the distributive property, you multiply the term outside the parentheses by each term inside the parentheses. For the expression \( b(3b + 5) \), we distribute the \( b \) as follows:

• Multiply \( b \) by \( 3b \), which gives us \( 3b^2 \).
• Multiply \( b \) by \( 5 \), resulting in \( 5b \).

Now, the expression looks like this: \( b^2 + 3b^2 + 5b \).This step sets the foundation for simplifying expressions by breaking them down and making them easier to manage.

Combining Like Terms

Combining like terms is crucial in simplifying polynomial expressions. Similar to sorting socks by color, we group terms with the same variable and power. This helps reduce an expression to its simplest form.
In our example:

• Combine the \( b^2 \) terms: \( b^2 + 3b^2 = 4b^2 \).
• Leave the \( 5b \) as it is, since there are no other \( b \) terms to combine with.

After combining, the expression becomes: \( 4b^2 + 5b \).There are no more like terms to combine, and the expression is in its simplest form. Remember that combining like terms not only simplifies the expression but can also provide clearer insight into what the expression represents.