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Chapter 5: Problem 64

Factor completely. See Examples 1 through \(9 .\) $$ \frac{a^{2}}{4}-\frac{b^{2}}{49} $$

Short Answer

Expert verified
The expression is factored as \( \left( \frac{a}{2} - \frac{b}{7} \right) \left( \frac{a}{2} + \frac{b}{7} \right) \).

Step by step solution

01

Rewriting the Fraction as a Difference of Squares

The expression given is \( \frac{a^2}{4} - \frac{b^2}{49} \). Recognize that both terms are squared fractions. They can be rewritten as a difference of squares: \( \left( \frac{a}{2} \right)^2 - \left( \frac{b}{7} \right)^2 \).
02

Applying the Difference of Squares Formula

Recall that the expression \( x^2 - y^2 \) can be factored as \( (x - y)(x + y) \). For this problem, substitute \( x = \frac{a}{2} \) and \( y = \frac{b}{7} \). This gives us: \[ \left( \frac{a}{2} - \frac{b}{7} \right) \left( \frac{a}{2} + \frac{b}{7} \right) \].
03

Arriving at the Final Factored Form

The completely factored expression for the original problem is \( \left( \frac{a}{2} - \frac{b}{7} \right) \left( \frac{a}{2} + \frac{b}{7} \right) \). Verify this by using the original difference of squares identity; it should return to the given expression.

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Key Concepts

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

Difference of Squares
The difference of squares is a specific algebraic pattern where two squared terms are subtracted from each other. It is written in the form \( a^2 - b^2 \). This pattern is significant because it allows us to factor the expression into two binomials, \((a - b)(a + b)\). Recognizing this pattern simplifies the process of solving or simplifying complex algebraic problems. In the provided exercise, we have \( \left( \frac{a}{2} \right)^2 - \left( \frac{b}{7} \right)^2 \). This corresponds directly to the difference of squares format, with \( x = \frac{a}{2} \) and \( y = \frac{b}{7} \). By identifying and applying this pattern, the expression can be effortlessly factored.
Algebraic Expressions
Algebraic expressions are combinations of numbers, letters (representing variables), and mathematical operations such as addition, subtraction, multiplication, and division. These expressions can be as simple as \(x + y\) or as complex as \(\frac{x^2}{y} - 3xy + 7\).
In our exercise, \(\frac{a^2}{4} - \frac{b^2}{49}\) is an algebraic expression involving two variables, \(a\) and \(b\), along with fractions. Understanding how to manipulate algebraic expressions, like recognizing patterns or simplifying terms, is crucial in algebra.
Factoring Techniques
Factoring is a crucial skill in algebra that involves rewriting an expression as a product of simpler expressions. Various techniques exist, each applicable to specific types of expressions. For instance:
  • Factoring out the Greatest Common Factor (GCF)
  • Factoring by grouping
  • Using special formulas like the difference of squares
In the given exercise, the difference of squares technique was used, which is ideal for expressions of the form \( a^2 - b^2 \).
When faced with such expressions, it's helpful to rewrite each term in squared form and apply the identity \( (x - y)(x + y) \). This strategic application of factoring techniques streamlines solving even complex algebraic equations.
Rational Expressions
Rational expressions are quotients of two polynomials. Simply put, they are fractions involving variables. Like regular fractions, simplifying rational expressions often involves factoring both the numerator and the denominator to cancel common factors.
In the example \( \left( \frac{a}{2} \right)^2 - \left( \frac{b}{7} \right)^2 \), each term is a fraction, but the expression itself isn't a rational expression until placed over another polynomial. Although not a direct example of simplifying a rational expression, this problem highlights the importance of recognizing patterns and factorization skills used when dealing with rational expressions in broader algebraic contexts.

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Most popular questions from this chapter

Factor each polynomial completely. See Examples 1 through 12. $$ (x-3)^{2}-2(x-3)-8 $$

Which numbers are equal to \(36,000 ?\) Of these, which is written in scientific notation? $$a. 36 \times 10^{3}$$ $$b.360 \times 10^{2}$$ $$c. 0.36 \times 10^{5}$$ $$d. 3.6 \times 10^{4}$$

Factor each polynomial completely. See Examples 1 through 12. $$ 4 x^{2}+12 x+9 $$

Find the value of \(c\) that makes each trinomial a perfect square trinomial. $$ n^{2}-2 n+c $$

If \(P(x)\) is the polynomial given, find a. \(P(a),\) b. \(P(-x),\) and c. \(P(x+h)\). \(P(x)=8 x+3\)
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