Problem 45 - A Graphical Approach to College Algebra

Question

Find each product. $$[(2 p-3)+q]^{2}$$

Step-by-Step Solution

Answer

\( (2p-3+q)^2 = 4p^2 - 12p + 9 + 4pq - 6q + q^2 \)

Step 1: Recognize the Binomial Squared

The expression \([(2p-3)+q]^2\) is a binomial squared, which can be expanded using the binomial formula \((a+b)^2 = a^2 + 2ab + b^2\). Here, let \(a = (2p-3)\) and \(b = q\).

Step 2: Calculate \(a^2\)

Substitute \(a = (2p-3)\) into the formula \(a^2\). Therefore, \((2p-3)^2 = (2p-3)(2p-3) = 4p^2 - 12p + 9\) using the distributive property (FOIL method).

Step 3: Calculate \(2ab\)

Calculate \(2ab = 2((2p-3)q) = 2(2pq - 3q) = 4pq - 6q\).

Step 4: Calculate \(b^2\)

Since \(b = q\), then \(b^2 = q^2\).

Step 5: Combine the Results

Add all the components from the previous steps to get the expanded form: \(4p^2 - 12p + 9 + 4pq - 6q + q^2\).

Step 6: Simplify the Expression

The expression \(4p^2 - 12p + 9 + 4pq - 6q + q^2\) is simplified as it does not have any like terms to combine.

Key Concepts

Polynomial Expansion

The concept of polynomial expansion is essential when dealing with algebraic expressions, particularly those that involve powers of a binomial. In our exercise, we are asked to find the product of \([(2p-3)+q]^2\). This requires us to expand this expression from its compact binomial form into a more detailed polynomial form. When you have a squared binomial, such as \[(a+b)^2\], the expansion follows the formula:

• \(a^2\)
• \(2ab\)
• \(b^2\)

Using this formula helps break down the expression into individual terms that can be further calculated. Each element in the formula represents a part of the expanded polynomial, making the operation systematic and straightforward. In our case:

• \(a=(2p-3)\)
• \(b=q\)

This structured approach allows students to handle complex algebraic expressions with ease, providing a clearer path to arrive at the final expanded form.

Distributive Property

The distributive property is a fundamental principle in algebra that permits us to break down expressions into simpler parts to make computation easier. It's often used when multiplying a single term by a sum or difference inside parentheses. In our specific example, \((2p-3)^2\), we use the distributive property via the FOIL method:

• First: Multiply the first terms \((2p)\times(2p)\)
• Outer: Multiply the outer terms \((2p)\times(-3)\)
• Inner: Multiply the inner terms \((-3)\times(2p)\)
• Last: Multiply the last terms \((-3)\times(-3)\)

This systematic approach helps simplify what might seem like a complicated multiplication into smaller, more manageable steps. Each step applies the distributive property to achieve parts of the full polynomial. In doing so, the original expression is decomposed into components that can be easily combined to find the expanded polynomial form.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They form the foundation of algebra, allowing us to represent mathematical relationships in a general form. In this exercise, the expression \([(2p-3)+q]^2\) is an algebraic expression that we expand into a polynomial. This involves integrating key algebraic concepts such as variables (\(p\) and \(q\)) and constants, and applying appropriate operations. To manage algebraic expressions effectively, it's crucial to understand how different parts interact:

• Variables represent unknown values and can change.
• Constants are fixed numbers.
• Operations such as addition, subtraction, multiplication, and division modify these quantities.

Understanding how algebraic expressions work is vital because they are the basis for writing equations and inequalities. This particular problem demonstrates how expanding an algebraic expression can clarify the structure and interrelations of its components, allowing us to see how different terms combine to form a polynomial.