Problem 61 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. $$p(4 p-6)+2(3 p-8)$$

Step-by-Step Solution

Answer

The simplified expression is \( 4p^2 - 16 \).

Step 1: Distribute the first term

First, distribute the expression within the first set of parentheses: \( p(4p - 6) \). Apply the distributive property: multiply \( p \) by each term inside the parentheses: \[p \times 4p = 4p^2 \quad \text{and} \quad p \times (-6) = -6p\]Thus, the expression becomes \( 4p^2 - 6p \).

Step 2: Distribute the second term

Next, distribute the expression within the second set of parentheses: \( 2(3p - 8) \). Multiply 2 by each term inside the parentheses: \[2 \times 3p = 6p \quad \text{and} \quad 2 \times (-8) = -16\]This results in \( 6p - 16 \).

Step 3: Combine and Simplify

Now, combine the results from Step 1 and Step 2:\[(4p^2 - 6p) + (6p - 16) = 4p^2 - 6p + 6p - 16\]Notice that \( -6p \) and \( 6p \) are like terms and will cancel each other out. This leaves:\[4p^2 - 16\]

Step 4: Final expression

The simplified final expression is \( 4p^2 - 16 \), which is the result of performing the given operations on the expression.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic rule that allows us to simplify expressions involving parentheses. It states that when multiplying a number, or even a variable, by a set of terms within parentheses, you can "distribute" the multiplication to each term inside. This property is essential for breaking down more complex expressions and making them easier to manage.
For example, if you have an expression like \( a(b + c) \), you can use the distributive property to rewrite it as \( ab + ac \).
In the original exercise, the expression \( p(4p - 6) \) was simplified by applying the distributive property:

• Multiply \( p \) by \( 4p \) to get \( 4p^2 \).
• Multiply \( p \) by \( -6 \) to get \( -6p \).

This process creates a new expression: \( 4p^2 - 6p \). Continuing in the same manner with the expression \( 2(3p - 8) \),

• Multiply \( 2 \) by \( 3p \) to get \( 6p \).
• Multiply \( 2 \) by \( -8 \) to obtain \( -16 \).

These simple multiplication tasks illustrate the power and usefulness of the distributive property.

Polynomial Operations

Polynomial operations are essential skills in algebra, involving the addition, subtraction, multiplication, and sometimes division of polynomials. A polynomial is an expression made up of variables, coefficients, and exponents.
When working with polynomial operations, we often come across the need to distribute terms, combine like terms, and simplify expressions. In the exercise, this is seen when dealing with the initial expression

• First distribution: \( p(4p - 6) \), resulting in \( 4p^2 - 6p \).
• Second distribution: \( 2(3p - 8) \), leading to \( 6p - 16 \).

Polynomial multiplication, as seen here, involves distributing each term across others, which leads to more complex expressions if there are more terms involved.
Understanding polynomial operations allows students to manage these expressions and can simplify expressions like this, leading to easier computation and interpretation of algebraic problems.

Simplifying Expressions

Simplifying expressions is a crucial step in making math problems easier to solve and understand. It involves reducing an expression to its simplest form by combining like terms, using basic operations, and applying algebraic rules effectively.
In our exercise, the expression ultimately needed simplifying to reach the final answer. After distributing and expanding all terms, we had the expression:

• \( 4p^2 - 6p + 6p - 16 \)

Noticing that \( -6p \) and \( 6p \) are like terms, we combined them. These terms are additive inverses, meaning they cancel each other out:

• This leaves the expression as \( 4p^2 - 16 \).

This simplification results in a cleaner, more manageable expression, making it easier to understand and work with in further calculations. Mastering simplification helps students approach all kinds of algebraic expressions with confidence and ease. It is a fundamental skill in solving more extensive and complex mathematical problems.