Problem 32 - College Algebra with Corequisite Support

Question

For the following exercises, multiply the binomials. \((9 a-4)(9 a+4)\)

Step-by-Step Solution

Answer

The product is \(81a^2 - 16\).

Step 1: Identify the Formula

The given expression is \((9a-4)(9a+4)\). Notice that this expression follows the form of a difference of squares: \((x-y)(x+y) = x^2 - y^2\). In this case, identify \(x = 9a\) and \(y = 4\).

Step 2: Apply the Difference of Squares Formula

Substitute \(x = 9a\) and \(y = 4\) into the difference of squares formula: \(x^2 - y^2\). This becomes \((9a)^2 - 4^2\).

Step 3: Simplify the Expression

Calculate \((9a)^2\) and \(4^2\): - \((9a)^2 = (9a) \times (9a) = 81a^2\)- \(4^2 = 16\)Thus, the expression \((9a)^2 - 4^2\) simplifies to \(81a^2 - 16\).

Step 4: Final Result

The product of the binomials \((9a-4)\) and \((9a+4)\) is \(81a^2 - 16\).

Key Concepts

Binomials

A binomial is a type of polynomial that consists of exactly two terms. These terms are typically joined by either a plus (+) or minus (-) sign. In the given problem, (9a - 4) and (9a + 4) are examples of binomials.

When working with binomials, it's important to recognize their structure. The standard form of a binomial is (ax + b). Here, '9a' represents one part of each term, and the constant number '4' is the other part.

Recognizing binomials is essential when applying special algebraic formulas like the difference of squares, which makes computation more efficient and eliminates the need for complex multiplication processes.

Binomials are used frequently in algebra, and understanding their properties helps in simplifying and solving polynomial expressions effectively.

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials together. Binomials are a special case of polynomials, and multiplying them is a common exercise in algebra. In the example with (9a - 4) and (9a + 4), we use the difference of squares formula to simplify the process.

When multiplying polynomials, each term of the first polynomial must be multiplied by each term of the second polynomial. This process is called distributive property or FOIL method. However, when binomials take the form of a difference of squares, there's a shortcut:

• The formula offers a quicker solution: offers a quicker solution: offers a quicker solution: offers a quicker solution: \((x-y)(x+y) = x^2 - y^2\).
• This formula shows that the middle terms cancel each other, simplifying the solution directly to the difference of the squares of the first terms and the last terms.
• This means you calculate 'the first square minus the second square' directly.

Recognizing these patterns allows for faster and easier solutions in polynomial multiplication.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that makes equations easier to understand and solve. In the case of multiplying binomials using the difference of squares, simplifying means recognizing and applying special formulas to shorten the calculation process.

For example, with (9a - 4)(9a + 4), the difference of squares formula means we have:

• Calculate the square of the first term: \((9a)^2 = 81a^2\).
• Calculate the square of the second term: \(4^2 = 16\).
• Subtract the second result from the first: \(81a^2 - 16\).

This simplification helps to quickly arrive at the solution 81a² - 16, showing how pre-existing patterns in algebra can simplify complex calculations.

Simplifying expressions using such formulas not only helps in quick problem-solving but also strengthens the understanding of algebraic identities, making it easier to handle more challenging problems later on.