Problem 33 - College Algebra with Corequisite Support

Question

For the following exercises, multiply the binomials. \((15 n-6)(15 n+6)\)

Step-by-Step Solution

Answer

225n^2 - 36

Step 1: Identify the Binomials

In the expression \((15n-6)(15n+6)\), notice the binomials are of the form \((a-b)(a+b)\), where \(a = 15n\) and \(b = 6\).

Step 2: Recall the Difference of Squares Formula

Recognize this expression as a difference of squares problem, which can be simplified using the formula: \((a-b)(a+b) = a^2 - b^2\).

Step 3: Apply the Difference of Squares Formula

Apply the formula: substitute \(a = 15n\) and \(b = 6\) into \(a^2 - b^2\). This gives \((15n)^2 - 6^2\).

Step 4: Calculate \(a^2\)

Calculate \((15n)^2\): it results in \((15n)^2 = 225n^2\).

Step 5: Calculate \(b^2\)

Calculate \(6^2\): it results in \(6^2 = 36\).

Step 6: Substitute and Simplify

Substitute \(a^2 = 225n^2\) and \(b^2 = 36\) into \(a^2 - b^2\). The expression becomes \(225n^2 - 36\).

Key Concepts

Binomials

A binomial is a type of polynomial that contains exactly two terms. Polynomials themselves are algebraic expressions made up of variables and coefficients, linked together by operations of addition, subtraction, and multiplication. When dealing with binomials, it's essential to note how the two terms interact with each other during operations such as multiplication.

A clear example of a binomial is the expression \(15n - 6\). Here, we have two separate terms: \(15n\) and \(-6\). Similarly, another binomial is \(15n + 6\), composed of \(15n\) and \(+6\). Each binomial can be expanded, simplified, or multiplied with others to form more complex expressions, which are often useful in solving algebraic equations.

When multiplying binomials, it's crucial to identify their terms correctly, as this identification helps in applying algebraic formulas effectively.

Difference of Squares Formula

The difference of squares formula is a specific method used in algebra to simplify the multiplication of binomials of the form \(a - b\) and \(a + b\). This formula states that \( (a-b)(a+b) = a^2 - b^2 \). It is an extremely efficient technique for quickly resolving expressions where terms have this form.

Let's analyze the example from the given exercise: \( (15n - 6)(15n + 6) \). Here, the expression fits the \(a - b\) and \(a + b\) structure, allowing us to directly use the difference of squares formula.

• Identify \(a\) as \(15n\) and \(b\) as \(6\).
• The formula then becomes \( (15n)^2 - (6)^2 \), which leads to further simplification.

By applying this formula, we have reduced the complexity of the expression to two simple calculations. This method not only saves time but also minimizes the chances of error in manual calculations.

Polynomial Multiplication

Polynomial multiplication extends the rules of arithmetic multiplication to expressions involving variables. When multiplying polynomials like binomials, it's important to maintain order and apply methods like the distributive property or special formulas, such as the difference of squares.

For the given exercise, polynomial multiplication simplifies with the difference of squares formula, reducing the calculation intensity from four distribution steps to only two main calculations:

• The term \( (15n)^2 \) corresponds to multiplying the term with itself, resulting in \( 225n^2 \).
• Similarly, \( (6)^2 \) results in \( 36 \).

After obtaining these results, we conclude the solution by performing a simple subtraction, yielding \((225n^2 - 36)\).

Understanding polynomial multiplication in this way not only helps grasp the particular solution but prepares students for tackling more complex algebraic expressions efficiently.