Problem 21 - Algebra 2 and Trigonometry

Question

Perform the indicated operations and write the result in simplest form. \(4 y(2 y-3)-5(2-y)\)

Step-by-Step Solution

Answer

The simplest form is \(8y^2 - 7y - 10\).

Step 1: Distribute Terms

First, distribute the terms in the expression: Multiply each term inside the parentheses by the term outside the parentheses. For the term \(4y(2y - 3)\): \(4y \times 2y = 8y^2\) and \(4y \times (-3) = -12y\). For the term \(-5(2-y)\): \(-5 \times 2 = -10\) and \(-5 \times (-y) = 5y\). This gives the expression: \(8y^2 - 12y - 10 + 5y\).

Step 2: Combine Like Terms

Next, combine like terms in the expression:Like terms are those with the same variable and exponent. In this expression:Combine \(-12y\) and \(5y\):\(-12y + 5y = -7y\).There are no other like terms with \(y^2\) or constant terms.Resulting expression: \(8y^2 - 7y - 10\).

Step 3: Write in Simplest Form

The expression \(8y^2 - 7y - 10\) is already in its simplest form since all like terms have been combined and there are no common factors. So, the expression remains as it is.

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that helps to multiply a single term across terms in parentheses. This property states that for any numbers or variables, if you have an expression in the form of \(a(b + c)\), it is equivalent to \(ab + ac\). This means you distribute or "spread" the outside term to every term inside the parentheses.

In our exercise, we start by distributing two separate terms. First, the expression \(4y(2y - 3)\) means you multiply \(4y\) with each term inside the parentheses. You get:

• \(4y \times 2y = 8y^2\)
• \(4y \times (-3) = -12y\)

Next, apply the distributive property to the second part, \(-5(2 - y)\):

• \(-5 \times 2 = -10\)
• \(-5 \times (-y) = 5y\). Notice multiplying two negatives gives a positive.

Once the operation is done, you create a new expression that will then be simplified further through other methods.

Combining Like Terms

Combining like terms is another key skill in algebra essential for simplifying expressions. Terms are "like" if they contain the same variable raised to the same power. Essentially, it’s the part of algebra where you "clean up" an expression by bringing together terms that have identical variables and exponents.

In the expression we worked on, after the distribution step, it is crucial to identify like terms. Here, the like terms to combine are the variable terms. Our expression is \(8y^2 - 12y - 10 + 5y\). The terms \(-12y\) and \(5y\) are like terms because they both contain the variable \(y\) to the first power.

• Combine these by performing the operation: \(-12y + 5y = -7y\)

After combining these, no other terms share the same variable and exponent, thus, no further combining is needed. As a result, we reach our simplified expression \(8y^2 - 7y - 10\). This expression is now much easier to understand and use in further calculations.

Simplifying Expressions

Simplifying an expression means reducing it to its most straightforward and concise form. In algebra, this means combining like terms and using arithmetic to present the expression in an easily understandable way.

After applying the distributive property and combining like terms, we ensure the expression is as simple as possible. The expression \(8y^2 - 7y - 10\) is now in its simplest form. But what makes it simple?

• There are no like terms left to combine.
• Each term is clearly separated and at its simplest form.
• No further factoring can be done since there are no common factors among all terms.

Having a simplified expression is crucial for solving equations, as it provides a clear view of the relationships between different terms, making further operations more manageable. Always aim for simplicity, as it helps in not only solving but also understanding and explaining the expression better.