Problem 94 - Beginning and Intermediate Algebra

Question

Find the following special products. $$ -(2 p+7 q)(2 p-7 q) $$

Step-by-Step Solution

Answer

The short answer to the given problem is: \[ -(2p + 7q)(2p - 7q) = -56pq \]

Step 1: Identify the expressions of a and b

In the given expression, we have to consider (2p + 7q) as 'a' and (2p - 7q) as 'b'. So, we can rewrite the given expression as (-1)(a)(b), where a = (2p + 7q) and b = (2p - 7q).

Step 2: Apply the formula (a+b)(a-b) = a² - b²

Now, we apply the formula (a+b)(a-b) = a² - b². Plugging in the previously identified terms a and b. \[ -(2p + 7q)(2p - 7q) = -( a^2 - b^2 ) \]

Step 3: Calculate a² and b²

We can compute a² = (2p + 7q)², and b² = (2p - 7q)². Let's perform the calculations: \[a^2 = (2p + 7q)^2 = (2p)^2 + 2(2p)(7q) + (7q)^2 \] \[a^2 = 4p^2 + 28pq + 49q^2 \] \[b^2 = (2p - 7q)^2 = (2p)^2 - 2(2p)(7q) + (7q)^2 \] \[b^2 = 4p^2 - 28pq + 49q^2 \]

Step 4: Calculate a² - b²

Now, perform the subtraction a² - b²: \[ a² - b² = (4p^2 + 28pq + 49q^2) - (4p^2 - 28pq + 49q^2) \]

Step 5: Simplify the expression

Simplify the expression by canceling out like terms: \[ a² - b² = 4p^2 + 28pq + 49q^2 - 4p^2 + 28pq - 49q^2 \] \[ a² - b² = 56pq \]

Step 6: Apply the negative sign

Don't forget to apply the negative sign from the beginning: \[ -( a² - b² ) = - 56pq \] So the product of the given expressions is: \[ -(2p + 7q)(2p - 7q) = - 56pq \]

Key Concepts

Difference of Squares

The concept of the difference of squares is a fundamental identity in algebra that helps to simplify expressions involving polynomial multiplication. It describes how the product of a sum and a difference of the same two terms can be simplified. In mathematical terms, this identity is expressed as:

• \((a+b)(a-b) = a^2 - b^2\)

This formula shows that when you multiply a binomial with its conjugate, you get the difference between the squares of the two terms. It's especially useful for quickly simplifying expressions and finding products without requiring individual term multiplication.

Algebraic Expression

Algebraic expressions are a way of representing numbers and operations using variables, constants, and algebraic operations like addition, subtraction, multiplication, and division. They can be as simple as a single number or variable, or as complex as a combination of several terms and operators.
For instance, in the given exercise, the algebraic expression is the product of two binomials:

• \(-(2p+7q)(2p-7q)\)

Understanding the structure and components of algebraic expressions is vital for simplifying and solving equations in algebra.

Polynomial Multiplication

Polynomial multiplication involves combining two polynomials to form another polynomial. When multiplying two binomials, each term in the first binomial must be multiplied by each term in the second.
In this exercise, however, the multiplication is simplified using the difference of squares formula. Instead of multiplying each term separately, we identify the pattern:

• \((2p+7q)(2p-7q)\)

This pattern leads directly to the result \(a^2 - b^2\), where \(a = 2p + 7q\) and \(b = 2p - 7q\), making the process faster and more efficient by skipping several steps.

Simplification in Algebra

Simplification in algebra refers to the process of making an algebraic expression as simple as possible. This involves reducing the complexity of the expression without changing its value.
In the exercise, after applying the difference of squares formula, the expression is:

• \(a^2 - b^2 = 56pq\)

From this step, simplification continues by ensuring all calculations are correctly performed and all like terms are combined or canceled out. Applying the negative sign at the end gives us the fully simplified expression:

• \(-56pq\)

This process demonstrates how algebraic simplification can transform a complex expression into a more manageable form, which is crucial for solving problems efficiently.