Problem 63 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. $$-y\left(y^{2}-4\right)+6 y^{2}(2 y-3)$$

Step-by-Step Solution

Answer

The result is \( 11y^3 - 18y^2 + 4y \).

Step 1: Distribute Into the First Expression

Let's start by distributing the \( -y \) across the expression inside the first parentheses \( y^{2} - 4 \). \(-y(y^2 - 4) = -y imes y^2 + -y imes (-4) = -y^3 + 4y\).

Step 2: Distribute Into the Second Expression

Next, distribute the \( 6y^2 \) across the expression inside the second parenthesis \( 2y - 3 \).\(6y^2(2y - 3) = 6y^2 imes 2y + 6y^2 imes (-3) = 12y^3 - 18y^2\).

Step 3: Combine Like Terms

Now, combine the results from the first and second distributions:\(-y^3 + 4y + 12y^3 - 18y^2\).Combine the \( y^3 \) terms: \(-y^3 + 12y^3 = 11y^3\).The expression becomes \( 11y^3 - 18y^2 + 4y \).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra used to simplify expressions and solve equations. It states that for any numbers or expressions, you can multiply a single term across a sum or difference within parentheses. For instance, for expression \( a(b + c) \), the distributive property allows you to multiply \( a \) by both \( b \) and \( c \), yielding \( ab + ac \).

Let's see it in action with our example: we began by distributing \( -y \) through \( (y^2 - 4) \). This means multiplying \( -y \) by each term inside the parentheses:

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

We do the same for the second part, \( 6y^2(2y - 3) \), resulting in:

• \( 6y^2 \times 2y = 12y^3 \)
• \( 6y^2 \times -3 = -18y^2 \)

This step transforms the equation into simpler components that we can manage and further simplify.

Combining Like Terms

Combining like terms is another essential operation in simplifying algebraic expressions. It involves grouping terms that have the same variables raised to the same power. This step reduces the complexity of the algebraic expression, making it more manageable.

Once you've used the distributive property in the exercise, you gather and combine like terms. In our example, after distributing, we have:

• \(-y^3 + 4y + 12y^3 - 18y^2\)

To combine them, look for terms with the same variables and exponents:

• The \( y^3 \) terms: \(-y^3 + 12y^3 = 11y^3 \)
• The \( y^2 \) term remains \(-18y^2\)
• The \( y \) term stays \( 4y \)

After combining, the expression simplifies to: \( 11y^3 - 18y^2 + 4y \). This process makes it easier to evaluate or further manipulate the expression if needed.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations like addition, subtraction, multiplication, and division. Understanding their structure and the methods we use to simplify them—such as the distributive property and combining like terms—is key to solving algebra problems.

In the given exercise, we began with the expression: \(-y(y^2 - 4) + 6y^2(2y - 3)\). This expression includes:

• Variables: Here, \( y \) represents unknown values.
• Coefficients: These are numbers multiplying the variables, such as \(-1, 6, 2,\) and \(-3\).
• Terms: Individual parts of the expression like \(-y^3\) and \(4y\).

Knowing the components of an algebraic expression helps in applying the distributive property and combining like terms. It allows you to manipulate the expression to solve equations or simply to understand the relationships between the components involved. With practice, these operations become intuitive and form the backbone of tackling more complex algebraic problems.