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Magazine Chapter 1: Problem 76
Factor the polynomial. $$ 64 x^{2}-36 y^{2} $$
Short Answer
Expert verified
The polynomial is factored as \((8x + 6y)(8x - 6y)\).
Step by step solution
01
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 \((a + b)(a - b)\).
02
Identify \(a^2\) and \(b^2\)
Rewrite the polynomial \(64x^2 - 36y^2\) as \((8x)^2 - (6y)^2\). Here, \(a = 8x\) and \(b = 6y\).
03
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:\((8x + 6y)(8x - 6y)\).
04
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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: \((8x + 6y)(8x - 6y)\).
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.
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.
