Problem 107 - Intermediate Algebra

Question

Perform the indicated operations. $$ (3 y+1)\left(2 y^{2}+3 y+2\right) $$

Step-by-Step Solution

Answer

The result of multiplying \((3y + 1)(2y^2 + 3y + 2)\) is \(6y^3 + 11y^2 + 9y + 2\).

Step 1: Apply the Distributive Property

To perform the multiplication of the two expressions \((3y + 1)(2y^2 + 3y + 2)\), we use the distributive property to expand it.First, distribute \(3y\) over each term inside the second parenthesis: \(3y(2y^2 + 3y + 2) = 3y \cdot 2y^2 + 3y \cdot 3y + 3y \cdot 2\).

Step 2: Simplify the First Distribution

Simplify the terms obtained by distributing \(3y\):- \(3y \cdot 2y^2 = 6y^3\)- \(3y \cdot 3y = 9y^2\)- \(3y \cdot 2 = 6y\)Thus, the expression from the first distribution is \(6y^3 + 9y^2 + 6y\).

Step 3: Distribute the Second Term

Now, distribute the \(1\) over each term in the parenthesis:- \(1 \cdot 2y^2 = 2y^2\)- \(1 \cdot 3y = 3y\)- \(1 \cdot 2 = 2\)This results in \(2y^2 + 3y + 2\).

Step 4: Combine and Collect Like Terms

Now combine the results from the two distributions:\(6y^3 + 9y^2 + 6y + 2y^2 + 3y + 2\).Next, combine like terms:- Combine \(9y^2 + 2y^2 = 11y^2\)- Combine \(6y + 3y = 9y\)Thus, the simplified expression is \(6y^3 + 11y^2 + 9y + 2\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us simplify expressions. It tells us how to multiply a single term across terms inside parentheses. Imagine distributing gifts from a common pot; each term outside the parentheses is like a giver who shares something with each term inside.

For instance, when we have \((3y + 1)(2y^2 + 3y + 2)\), we need to apply the distributive property twice:

• First Distribution: Multiply \(3y\) with each term inside \((2y^2 + 3y + 2)\), resulting in \(3y \cdot 2y^2 + 3y \cdot 3y + 3y \cdot 2\).
• Second Distribution: Do the same for \(1\), multiplying it by each term inside the parentheses, giving us \(1 \cdot 2y^2 + 1 \cdot 3y + 1 \cdot 2\).

In both cases, we expand what's inside the parentheses, allowing us to eliminate the brackets and simplify further.

Like Terms

In algebra, like terms are those that have the same variable raised to the same power. They are like identical types of fruit that can be grouped together when simplifying expressions.

In the expression obtained after using the distributive property:\(6y^3 + 9y^2 + 6y + 2y^2 + 3y + 2\), identify the like terms:

• \(9y^2\) and \(2y^2\) both have \(y^2\) as their key ingredient.
• \(6y\) and \(3y\) both have a \(y\).

By grouping these, we can combine them:

• Add terms with \(y^2\): \(9y^2 + 2y^2 = 11y^2\).
• Add terms with \(y\): \(6y + 3y = 9y\).

Recognizing and combining like terms simplifies polynomial expressions significantly.

Simplifying Expressions

Simplifying expressions is one of the key steps in solving algebraic problems. It involves reducing expressions to their simplest form.

Once we apply the distributive property and group like terms, our goal is to simplify:
Start from an expanded form, \(6y^3 + 9y^2 + 6y + 2y^2 + 3y + 2\),
and use the combinations of like terms to reach a simplified form: \(6y^3 + 11y^2 + 9y + 2\).

• Key Steps in Simplification:
• Always perform distribution properly to get rid of parentheses.
• Combine like terms to merge terms with the same variable and exponent.
• Rewrite the expression in the simplest way possible for ease of calculation and interpretation.

Simplifying not only makes expressions less complicated, but it helps in understanding their structure and how they behave in equations or graphs.