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Chapter 6: Problem 38

Factor each binomial completely. \(x^{2}-225 y^{2}\)

Short Answer

Expert verified
The binomial factors to \((x - 15y)(x + 15y)\).

Step by step solution

01

Recognize the Difference of Squares

The expression to factor is \(x^2 - 225y^2\). Recognize that both \(x^2\) and \(225y^2\) are perfect squares. Recall the difference of squares identity: \(a^2 - b^2 = (a-b)(a+b)\).
02

Identify the Values for 'a' and 'b'

Identify \(a\) and \(b\) in the given expression. Here, \(a^2 = x^2\) gives \(a = x\), and \(b^2 = 225y^2\) gives \(b = 15y\).
03

Apply the Difference of Squares Formula

Substitute \(a\) and \(b\) into the difference of squares formula. This results in \((x - 15y)(x + 15y)\).
04

Verify the Factored Expression

Multiply \((x - 15y)\) by \((x + 15y)\) to ensure the expression is equivalently \(x^2 - 225y^2\). This confirms 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
In algebra, the "difference of squares" is a common factorization technique. It applies when you have an expression composed of two perfect squares separated by a subtraction sign. The general formula for the difference of squares is:
  • \(a^2 - b^2 = (a-b)(a+b) \)
This formula allows you to factor the expression into two linear binomials. It helps simplify calculations and solve equations more efficiently.
Whenever you spot a subtraction between two squared terms, this method can be extremely useful.
In our specific example, the expression \(x^2 - 225 y^2\) fits perfectly into the difference of squares criteria. Recognizing this pattern is critical to correctly factor the given expression.
Binomial
A "binomial" is a polynomial expression that consists of exactly two terms. These terms are usually connected by either a plus or minus sign. In the context of the difference of squares, each of the resulting factors is a binomial.
For instance, in the expression \((x - 15y)(x + 15y)\), both \(x - 15y\) and \(x + 15y\) are binomials.
Binomials play an essential role in factorization exercises since they often serve as building blocks for forming and manipulating expressions.
  • Recognizing and handling binomials helps in expanding expressions and simplifying complex algebraic fractions.
  • Practice identifying binomials to enhance your problem-solving agility, especially in algebraic manipulations.
Perfect Squares
"Perfect squares" are numbers or expressions that can be expressed as the square of a whole number or another polynomial. Recognizing perfect squares is key when applying the difference of squares because these expressions are precisely what you need to factor.
For example, in the expression \(x^2 - 225y^2\), both \(x^2\) and \(225y^2\) are perfect squares.
  • \(x^2\) simplifies to \(x\times x\) and \(225y^2\) simplifies to \((15y)\times (15y)\).
It is crucial to identify the square root of each term easily. This will allow you to extract the values necessary to construct your binomials.
Understanding perfect squares also aids in various other mathematical contexts, such as solving quadratic equations and simplifying square roots.

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Most popular questions from this chapter

Find a positive value of \(c\) so that each trinomial is factorable. $$ y^{2}-4 y+c $$

Factor out the GCF from each polynomial. See Examples 4 through 10. $$ 6 x^{3}-9 x^{2}+12 x $$

Factor each trinomial completely. $$ x^{2}+x+\frac{1}{4} $$

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1 ). Don't forget to factor out the GCF first. See Examples I through 10. $$ x^{3} y^{2}+x^{2} y-20 x $$

Factor out the GCF from each polynomial. See Examples 4 through 10. $$ 18 a+12 $$
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