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Chapter 0: Problem 64

Factor the expression completely, if possible. \(9 x^{2}-4 y^{2}\)

Short Answer

Expert verified
The expression factors to \((3x - 2y)(3x + 2y)\).

Step by step solution

01

Recognize the Form

Notice that the expression is of the form \(a^2 - b^2\), which is a difference of squares. The difference of squares can be factored using the identity \(a^2 - b^2 = (a-b)(a+b)\).
02

Identify \(a\) and \(b\)

Rewrite the expression recognizing the squares: \(9x^2 - 4y^2\). Here, \(9x^2\) is \((3x)^2\) and \(4y^2\) is \((2y)^2\). Hence, \(a = 3x\) and \(b = 2y\).
03

Apply the Difference of Squares Formula

Now that we have \(a = 3x\) and \(b = 2y\), substitute into the difference of squares formula: \((3x - 2y)(3x + 2y)\).
04

Verify the Factored Form

Multiply \((3x - 2y)\) and \((3x + 2y)\) to ensure it equals \(9x^2 - 4y^2\). Use the distributive property: \( (3x - 2y)(3x + 2y) = (3x)^2 + 3x(2y) - 2y(3x) - (2y)^2 = 9x^2 - 4y^2 \).The expression simplifies back to the original, confirming that the factorization is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares
A difference of squares is a specific type of algebraic expression where two perfect squares are subtracted from each other. In algebra,
  • "Difference" means subtraction, and
  • "Squares" refers to numbers or expressions that can be written as something multiplied by itself.

The recognizable formula for factoring a difference of squares is:\[ a^2 - b^2 = (a - b)(a + b) \]
Here, \( a \) and \( b \) are expressions that are squared. Spotting this form quickly simplifies the process of factorization. In our original problem, once we see the pattern \( 9x^2 - 4y^2 \), we rewrite it to show the squares, \((3x)^2 - (2y)^2\), allowing us to implement the difference of squares formula to factor the expression effectively.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operational symbols. It can be as simple as a single number or a letter representing an unknown value. In more complex expressions, like polynomials, they show relationships between quantities.
Some key things to remember about algebraic expressions are:
  • They can range from simple terms to larger polynomials.
  • Operations like addition, subtraction, multiplication, and division are used to build and manipulate them.
  • Variables represent values that can change or are unknown.

Understanding algebraic expressions is crucial because they form the building blocks of algebra. In the expression \(9x^2 - 4y^2\), the terms \(9x^2\) and \(4y^2\) are algebraic expressions involving multiplication of variables and numbers.
Polynomial Factorization
Polynomial factorization involves expressing a polynomial as a product of its factors, which are simpler polynomials. It's akin to breaking down a composite number into its prime factors, but with variables involved.
For successful factorization, consider the following steps:
  • Identify any common factors before delving into special formulas like difference of squares.
  • Recognize patterns such as the difference of squares, perfect square trinomials, or cubes.
  • Apply appropriate formulas or methods for each recognizable pattern.

In our case, the expression \(9x^2 - 4y^2\) was factored using the difference of squares identity, splitting it into \((3x - 2y)(3x + 2y)\). Proper factorization greatly simplifies solving equations and finding roots, which is a primary purpose in algebra.

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Most popular questions from this chapter

Apply the distributive property. $$-4(5 x-y)$$

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Exercises \(65-90:\) Use rules of exponents to simplify the expression. Use positive exponents to write your answer. $$ \left(\frac{-2 t}{4 t^{-2}}\right)^{-1} $$

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