Problem 76 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 64 x^{2}-36 y^{2} $$

Step-by-Step Solution

Answer

The polynomial is factored as \((8x + 6y)(8x - 6y)\).

Step 1: Recognize the Polynomial Form

The polynomial given is \(64x^2 - 36y^2\). Notice that this expression is a difference of squares. Recall that any expression of the form \(a^2 - b^2\) can be factored into \((a + b)(a - b)\).

Step 2: Identify \(a^2\) and \(b^2\)

Rewrite the polynomial \(64x^2 - 36y^2\) as \((8x)^2 - (6y)^2\). Here, \(a = 8x\) and \(b = 6y\).

Step 3: Apply the Difference of Squares Formula

Using the identity \(a^2 - b^2 = (a + b)(a - b)\), substitute \(a = 8x\) and \(b = 6y\) into the formula. This gives you the factors:\((8x + 6y)(8x - 6y)\).

Step 4: Simplify the Factors, If Possible

Check if the factors \((8x + 6y)(8x - 6y)\) can be simplified further. In this case, since 8x and 6y have no common factors with each other, this is the simplest form.

Key Concepts

Difference of Squares

In algebra, the difference of squares is a specific method used for factoring certain polynomial expressions. It applies when you have a subtraction (-) between two terms, each of which is a perfect square. This is identified by recognizing that the expression can fit the formula \(a^2 - b^2 = (a + b)(a - b)\). The key is to determine what values, when squared, give the terms of the polynomial.

• For example, in the polynomial \(64x^2 - 36y^2\), the terms \(64x^2\) and \(36y^2\) are both perfect squares.
• We express these as \((8x)^2\) and \((6y)^2\), thereby identifying \(a = 8x\) and \(b = 6y\).
• This allows us to rewrite the polynomial using the difference of squares formula to get the factored form: \((8x + 6y)(8x - 6y)\).

Recognizing a difference of squares is a powerful tool because it simplifies the polynomial into a product of binomials, which are often easier to work with in algebraic manipulations.

Factoring Techniques

Factoring is a fundamental skill in algebra used to simplify expressions, solve equations or simply put polynomial expressions in a more manageable form. Multiple techniques exist for factoring, and choosing the right one depends on recognizing patterns in the expression.

• One common technique involves identifying common factors, known as common factor extraction, which simplifies expressions by dividing each term by the greatest common factor.
• Another technique is identifying special polynomial forms like the "difference of squares", which we've previously explored.
• More complex forms might use the "trinomial squares" or "grouping" method.

When faced with a polynomial, you should first look for these patterns or structures, as they guide you in choosing the most efficient factoring method. For the polynomial \(64x^2 - 36y^2\), using the difference of squares method is effective, leading directly to the factors \((8x + 6y)(8x - 6y)\). Recognizing these structures becomes easier with practice.

Polynomial Expressions

A polynomial is a mathematical expression that consists of variables, coefficients, and exponents, linked together by the operations of addition, subtraction, or multiplication. They play a crucial role in various areas of algebra.

• Polynomials are classified by their degree, which is determined by the highest exponent of the variable. For instance, in \(64x^2\), the degree is 2.
• Each term in a polynomial is made up of a coefficient and a variable raised to a power. In our example, \(64\) is the coefficient, and \(x^2\) is the term involving the variable.
• Factoring polynomials is a technique used to break them down into simpler components that are easier to work with.

Working with polynomial expressions often requires identifying different algebraic structures and applying correct or suitable factoring methods to simplify them. For \(64x^2 - 36y^2\), recognizing it as a difference of squares allowed us to factor it efficiently. Understanding these expressions' structure and operations applied to them is essential for mastering algebra.