Problem 51 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. $$(6 p+5 q)(3 p-7 q)$$

Step-by-Step Solution

Answer

\(18p^2 - 27pq - 35q^2\)

Step 1: Distribute the First Term

Start by distributing the first term of the first binomial, which is \(6p\), to each term in the second binomial \((3p - 7q)\). \(6p(3p) + 6p(-7q) = 18p^2 - 42pq\).

Step 2: Distribute the Second Term

Next, distribute the second term of the first binomial, which is \(5q\), to each term in the second binomial \((3p - 7q)\). \(5q(3p) + 5q(-7q) = 15pq - 35q^2\).

Step 3: Combine Like Terms

Combine the results from Step 1 and Step 2. Combine the \(pq\) terms: \(18p^2 - 42pq + 15pq - 35q^2\). This simplifies to: \(18p^2 - 27pq - 35q^2\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term across terms within a parenthesis. It's like spreading out or distributing the term to each part of the expression. In our problem, we have two binomials:

• The first one is \(6p + 5q\).
• The second one is \(3p - 7q\).

To apply the distributive property, multiply each term in the first binomial by each term in the second binomial:
1. Distribute \(6p\) to both \(3p\) and \(-7q\).
2. Distribute \(5q\) to both \(3p\) and \(-7q\).This results in:

• \(6p \times 3p = 18p^2\)
• \(6p \times -7q = -42pq\)
• \(5q \times 3p = 15pq\)
• \(5q \times -7q = -35q^2\)

The key to the distributive property is remembering to keep track of all terms and their signs.

Binomial Expansion

Binomial expansion is a process of expanding expressions that involve two terms raised to a power or multiplied together, like in our binomial multiplication example. When we multiply two binomials, each term in the first is distributed over all the terms in the second, giving us four products:- The product of the first terms: \(6p \times 3p = 18p^2\). - The product of the outer terms: \(6p \times -7q = -42pq\).- The product of the inner terms: \(5q \times 3p = 15pq\). - The product of the last terms: \(5q \times -7q = -35q^2\). Through binomial expansion, we see how each part of the expression fits together to form the complete expanded form. Essentially, it combines multiple small multiplications to construct a larger expression that is equivalent to multiplying the two binomials directly.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions, especially after expanding binomials. Like terms are terms that involve the same variables raised to the same power. In our example, after using the distributive property and expanding the binomials, we need to simplify:- Start with the expanded terms: \(18p^2 - 42pq + 15pq - 35q^2\).- Identify like terms, which in this case are the terms involving \(pq\): - \(-42pq\) and \(15pq\).To combine these, add or subtract the coefficients:\(-42pq + 15pq = -27pq\)So, the terms simplify to:\(18p^2 - 27pq - 35q^2\).This step is crucial because it reduces the expression to its simplest form, making it easier to interpret or use in further calculations.