Problem 47 - A Graphical Approach to College Algebra

Question

Find each product. $$[(3 q+5)-p][(3 q+5)+p]$$

Step-by-Step Solution

Answer

The product is \(9q^2 + 30q + 25 - p^2\).

Step 1: Recognize the Expression Format

The expression is in the format \((a - b)(a + b)\) which fits the difference of squares formula. Here, \(a = 3q + 5\) and \(b = p\).

Step 2: Apply the Difference of Squares Formula

The difference of squares formula states: \((a - b)(a + b) = a^2 - b^2\). Applying it here: \(((3q + 5) - p)((3q + 5) + p) = (3q + 5)^2 - p^2\).

Step 3: Calculate \((3q + 5)^2\)

Expand the square: \((3q + 5)^2 = (3q)^2 + 2(3q)(5) + 5^2 = 9q^2 + 30q + 25\).

Step 4: Substitute Back

Place the expanded terms into the formula: \((3q + 5)^2 - p^2 = 9q^2 + 30q + 25 - p^2\).

Step 5: Final Expression

The expression simplifies to \(9q^2 + 30q + 25 - p^2\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. The expression in our exercise involves combining these elements using both addition and subtraction. Here, we see a combination of variables like \(q\) and \(p\), with numbers such as 3 and 5. When handling algebraic expressions, it's crucial to understand the parts involved:• **Variables**: Symbols like \(q\) and \(p\) that represent unknown quantities. They allow for generalization in mathematics, which is a powerful tool for creating formulas and solving equations. • **Constants**: Numbers like 3 and 5 that are fixed values within an expression. They provide specific numerical characteristics to the algebraic expression. The given expression \([(3q + 5) - p][(3q + 5) + p]\) showcases a structure that is ripe for applying algebraic principles like the difference of squares, as it expresses a combination of terms that can be neatly simplified. Understanding each part of the expression is the first step in managing algebraic tasks effectively.

Introduction to Polynomial Expansion

Polynomial expansion involves multiplying out expressions to get a sum of terms. It is an essential concept in algebra since it provides clarity and simplification. In our exercise, we applied polynomial expansion to \((3q + 5)^2\).To expand a polynomial such as \( (a + b)^2 \), we apply the formula:\[ a^2 + 2ab + b^2 \].Here, we replace \(a\) with \(3q\) and \(b\) with 5, giving us:• \( (3q)^2 = 9q^2 \)• \( 2 \times 3q \times 5 = 30q \)• \( 5^2 = 25 \)This transforms \((3q + 5)^2\) into \(9q^2 + 30q + 25\). Polynomial expansions are powerful for simplifying expressions by distributing terms and using arithmetic rules. This expands any polynomial expression into an easy-to-read linear combination.

Utilizing Factoring Techniques

Factoring techniques are foundational tools in algebra that simplify polynomial expressions by expressing them as products of their factors. They make expressions easier to manipulate and solve, as seen in our exercise.For the given problem, \([(3q + 5) - p][(3q + 5) + p]\), we used the difference of squares factoring technique. This technique is based on the identity \[ (a - b)(a + b) = a^2 - b^2 \].In this identity, rather than multiplying out each term, we effortlessly factor to reach a straightforward expression.• **Recognize the Pattern**: Match the expression pattern to \((a - b)(a + b)\),identifying \(a = 3q + 5\) and \(b = p\).• **Application**: Applying the identity simplifies our expression to \((3q + 5)^2 - p^2\).Ultimately, using this technique reduces the workload involved in handling complex algebraic expressions, streamlining the process and ensuring accuracy with less effort. Factoring techniques like this make otherwise daunting algebraic problems manageable and logical.