Problem 54
Question
Perform the indicated operations. When possible write down only the answer. $$\left(x^{2}-y^{2}\right) \div\left(y^{2}-x^{2}\right)$$
Step-by-Step Solution
Verified Answer
The answer is \-1 \.
1Step 1 - Recognize the expression
Identify that \((x^{2} - y^{2}) \div (y^{2} - x^{2})\) involves the difference of squares in both numerator and denominator.
2Step 2 - Simplify the numerator
Factorize the numerator: \[ x^{2} - y^{2} = (x+y)(x-y) \]
3Step 3 - Simplify the denominator
Factorize the denominator: \[ y^{2} - x^{2} = (y+x)(y-x) \]
4Step 4 - Recognize the equivalence
Notice that \(y^{2} - x^{2} = -(x^{2} - y^{2})\). Therefore, \((y+x)(y-x) = -(x+y)(x-y)\).
5Step 5 - Divide the expressions
Divide the factored forms: \[ \frac{(x+y)(x-y)}{-(x+y)(x-y)} \]
6Step 6 - Simplify the expression
Cancel the common terms from numerator and denominator: \[(x+y)(x-y) \div (-(x+y)(x-y)) = -1 \]
Key Concepts
Difference of squaresFactoringSimplifying expressions
Difference of squares
The difference of squares is a common algebraic identity used to simplify mathematical expressions involving squares. It states that: \(a^2 - b^2 = (a + b)(a - b)\).
This identity is useful because it breaks down a squared term into a product of two simpler binomials.
Understanding this allows you to recognize that both \(x^2 - y^2\) and \(y^2 - x^2\) can be factorized.
In our example, we deal with \(x^2 - y^2\), which can be factorized into \((x+y)(x-y)\).
Similarly, \(y^2 - x^2\) is just the negative form of \(x^2 - y^2\), which can be expressed as \(-(x+y)(x-y)\).
This identity is useful because it breaks down a squared term into a product of two simpler binomials.
Understanding this allows you to recognize that both \(x^2 - y^2\) and \(y^2 - x^2\) can be factorized.
In our example, we deal with \(x^2 - y^2\), which can be factorized into \((x+y)(x-y)\).
Similarly, \(y^2 - x^2\) is just the negative form of \(x^2 - y^2\), which can be expressed as \(-(x+y)(x-y)\).
Factoring
Factoring involves breaking down an expression into simpler pieces that, when multiplied together, give back the original expression.
This is especially helpful with polynomials.
In our example, factoring transforms \(x^2 - y^2\) into \((x+y)(x-y)\) and \(y^2 - x^2\) into \((y+x)(y-x)\).
Recognizing these transformations is essential for simplifying the given expression.
By factoring the numerator and the denominator, we see that they share common terms, which will simplify our calculations significantly.
It's important to remember that factoring reveals structures within expressions that are otherwise hard to see.
This is especially helpful with polynomials.
In our example, factoring transforms \(x^2 - y^2\) into \((x+y)(x-y)\) and \(y^2 - x^2\) into \((y+x)(y-x)\).
Recognizing these transformations is essential for simplifying the given expression.
By factoring the numerator and the denominator, we see that they share common terms, which will simplify our calculations significantly.
It's important to remember that factoring reveals structures within expressions that are otherwise hard to see.
Simplifying expressions
Simplifying expressions involves reducing them to their most basic form.
After factoring both parts of our expression, we proceed to cancel out common terms.
For our expression \((x^2 - y^2) / (y^2 - x^2)\), both the numerator and the denominator factor to \((x+y)(x-y)\) and \(-(x+y)(x-y)\), respectively.
This allows us to divide and cancel these terms:
The negative sign comes from the difference in the order of terms, where \(y^2 - x^2 = -(x^2 - y^2)\).
By recognizing common factors and simplifying, we make the expression easier to handle and understand.
After factoring both parts of our expression, we proceed to cancel out common terms.
For our expression \((x^2 - y^2) / (y^2 - x^2)\), both the numerator and the denominator factor to \((x+y)(x-y)\) and \(-(x+y)(x-y)\), respectively.
This allows us to divide and cancel these terms:
- Numerator: \((x+y)(x-y)\)
- Denominator: \(-(x+y)(x-y))\)
The negative sign comes from the difference in the order of terms, where \(y^2 - x^2 = -(x^2 - y^2)\).
By recognizing common factors and simplifying, we make the expression easier to handle and understand.
Other exercises in this chapter
Problem 53
Simplify. $$\frac{a^{-1}-b^{-1}}{a-b}$$
View solution Problem 54
Solve each equation. $$\frac{6}{a+2}=\frac{a}{a+2}$$
View solution Problem 54
Solve each problem. Rodent food. A pest control specialist has found that 6 rats can eat an entire box of sugar-coated breakfast cereal in 13.6 minutes, and it
View solution Problem 55
Solve each equation. $$\frac{w}{6}=\frac{3}{2 w}$$
View solution