Problem 79
Question
Factor each numerator and denominator. Then simplify if possible. $$ \frac{7 x-7 y}{x^{2}-y^{2}} $$
Step-by-Step Solution
Verified Answer
\(\frac{7}{x+y}\)
1Step 1: Factor the Numerator
The numerator of the given fraction is \(7x - 7y\). Notice that both terms in the numerator have a common factor of 7. Factoring out the 7 gives us:\[7x - 7y = 7(x - y).\]
2Step 2: Factor the Denominator
The denominator of the given fraction is \(x^2 - y^2\). This expression is a difference of squares and can be factored using the identity \(a^2 - b^2 = (a - b)(a + b)\). Applying this identity, we have:\[x^2 - y^2 = (x - y)(x + y).\]
3Step 3: Write the Fraction with Factored Expressions
Substitute the factored forms of the numerator and the denominator into the original fraction:\[\frac{7(x - y)}{(x - y)(x + y)}.\]
4Step 4: Simplify the Fraction by Canceling Common Factors
Both the numerator and the denominator have a common factor of \((x - y)\). Cancel this common factor to simplify the fraction:\[\frac{7(x - y)}{(x - y)(x + y)} = \frac{7}{x + y}.\]
5Step 5: Conclusion: Simplified Form
The simplified form of the given expression is \(\frac{7}{x + y}\). This is the final answer after factoring and simplifying the given fraction.
Key Concepts
Difference of SquaresCommon FactorRational ExpressionsSimplification of Fractions
Difference of Squares
In algebra, one common type of expression you will encounter is the difference of squares. This involves two terms, each being a perfect square, separated by a subtraction sign.
The general form is given by the expression \( a^2 - b^2 \), which can be factored into \( (a-b)(a+b) \).
This identity is a useful tool in algebra, as it simplifies complex expressions quickly and easily. For example, in the given exercise, the denominator is \( x^2 - y^2 \). Recognizing this as a difference of squares allows us to rewrite it as \( (x-y)(x+y) \).
This factoring method helps in simplifying algebraic fractions, making expressions more manageable and easier to work with in further mathematical operations.
The general form is given by the expression \( a^2 - b^2 \), which can be factored into \( (a-b)(a+b) \).
This identity is a useful tool in algebra, as it simplifies complex expressions quickly and easily. For example, in the given exercise, the denominator is \( x^2 - y^2 \). Recognizing this as a difference of squares allows us to rewrite it as \( (x-y)(x+y) \).
This factoring method helps in simplifying algebraic fractions, making expressions more manageable and easier to work with in further mathematical operations.
Common Factor
Identifying common factors is a fundamental step in simplifying algebraic expressions. A common factor is a term that appears in every part of an expression.
By factoring it out, we break down the expression into a simpler form, which can often be more easily manipulated.In the case of the numerator \( 7x - 7y \) from the original exercise, the common factor is 7.
Factoring out this 7, we get \( 7(x - y) \). This not only reveals the structure of the expression but also prepares it for further simplification in the context of a fraction.Here’s why this step is crucial:
By factoring it out, we break down the expression into a simpler form, which can often be more easily manipulated.In the case of the numerator \( 7x - 7y \) from the original exercise, the common factor is 7.
Factoring out this 7, we get \( 7(x - y) \). This not only reveals the structure of the expression but also prepares it for further simplification in the context of a fraction.Here’s why this step is crucial:
- It reduces the complexity of an expression, making it easier to simplify further.
- It often reveals additional factors that may lead to further simplification.
- Factoring helps in canceling terms in rational expressions, which is essential for simplification.
Rational Expressions
Rational expressions are fractions where both the numerator and the denominator are polynomials.
These can often be simplified by factoring. Simplifying rational expressions is useful because it reduces them to their simplest form, making them easier to interpret and use.In our exercise, the expression is \( \frac{7x - 7y}{x^2 - y^2} \).
Each part of the expression is a polynomial: the numerator is a linear polynomial, and the denominator is quadratic.
The process of simplifying these involves factoring both the numerator and the denominator and then canceling any common factors.The goal with rational expressions is to:
These can often be simplified by factoring. Simplifying rational expressions is useful because it reduces them to their simplest form, making them easier to interpret and use.In our exercise, the expression is \( \frac{7x - 7y}{x^2 - y^2} \).
Each part of the expression is a polynomial: the numerator is a linear polynomial, and the denominator is quadratic.
The process of simplifying these involves factoring both the numerator and the denominator and then canceling any common factors.The goal with rational expressions is to:
- Identify any possible simplifications.
- Factor polynomials as much as possible.
- Cancel common factors where permitted.
Simplification of Fractions
Simplification of fractions involves reducing the fraction to its simplest form so that the numerator and denominator have no common factors, other than 1. It makes expressions more elegant and easier to handle in equations or additional operations.In the given exercise, after factoring both the numerator and the denominator, the fraction simplifies to \( \frac{7(x-y)}{(x-y)(x+y)} \).
The common factor \( (x-y) \) is then canceled out, leaving us with \( \frac{7}{x+y} \). Here's why simplification is important:
The common factor \( (x-y) \) is then canceled out, leaving us with \( \frac{7}{x+y} \). Here's why simplification is important:
- It reduces complexity, making it easier to evaluate and solve expressions.
- It highlights key features of the expression, such as critical points in rational expressions.
- Simplified forms are often required in further mathematical operations, such as integration or limits in calculus.
Other exercises in this chapter
Problem 79
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