Get expert advice and insights on any topic with IDNLearn.com. Ask anything and receive prompt, well-informed answers from our community of knowledgeable experts.
Sagot :
To solve the expression [tex]\(\frac{x^2 - 4y^2}{x + y} \times \frac{x^2 - xy}{2x^2 + 5xy + 2y^2} \div \frac{x^2 y - y^2}{y + 2x}\)[/tex], follow these steps:
### Step 1: Simplify each fraction separately.
1. Simplify [tex]\(\frac{x^2 - 4y^2}{x + y}\)[/tex]
Notice that [tex]\(x^2 - 4y^2\)[/tex] is a difference of squares:
[tex]\[ x^2 - 4y^2 = (x - 2y)(x + 2y) \][/tex]
Therefore,
[tex]\[ \frac{x^2 - 4y^2}{x + y} = \frac{(x - 2y)(x + 2y)}{x + y} \][/tex]
This simplifies to:
[tex]\[ x - 2y \quad \text{if} \quad x + y \neq 0 \][/tex]
2. Simplify [tex]\(\frac{x^2 - xy}{2x^2 + 5xy + 2y^2}\)[/tex]
The expression [tex]\(x^2 - xy\)[/tex] can be factored as:
[tex]\[ x^2 - xy = x(x - y) \][/tex]
We factor the denominator [tex]\(2x^2 + 5xy + 2y^2\)[/tex]. This can be recognized as a quadratic trinomial which factors as:
[tex]\[ 2x^2 + 5xy + 2y^2 = (2x + y)(x + 2y) \][/tex]
Therefore,
[tex]\[ \frac{x^2 - xy}{2x^2 + 5xy + 2y^2} = \frac{x(x - y)}{(2x + y)(x + 2y)} \][/tex]
3. Simplify [tex]\(\frac{x^2 y - y^2}{y + 2x}\)[/tex]
Factor the numerator [tex]\(x^2 y - y^2\)[/tex]:
[tex]\[ x^2 y - y^2 = y(x^2 - y) = y(x + y)(x - y) \][/tex]
Thus,
[tex]\[ \frac{x^2 y - y^2}{y + 2x} = \frac{y(x + y)(x - y)}{y + 2x} \][/tex]
### Step 2: Perform the division and multiplication.
Since division by a fraction is equivalent to multiplication by its reciprocal, we rewrite the operation:
[tex]\[ \frac{x^2 - 4y^2}{x + y} \times \frac{x^2 - xy}{2x^2 + 5xy + 2y^2} \div \frac{x^2 y - y^2}{y + 2x} = \frac{x^2 - 4y^2}{x + y} \times \frac{x^2 - xy}{2x^2 + 5xy + 2y^2} \times \frac{y + 2x}{x^2 y - y^2} \][/tex]
Substitute the simplified forms we obtained:
[tex]\[ (x - 2y) \times \frac{x(x - y)}{(2x + y)(x + 2y)} \times \frac{y + 2x}{y(x - y)} \][/tex]
### Step 3: Simplify the resulting expression.
1. Combine similar terms and cancel:
[tex]\[ = (x - 2y) \times \frac{x(x - y)}{(2x + y)(x + 2y)} \times \frac{y + 2x}{y(x - y)} \][/tex]
2. Notice that [tex]\( (x - y) \)[/tex] appears in both numerator and denominator, so they cancel:
[tex]\[ = (x - 2y) \times \frac{x}{(2x + y)(x + 2y) \times y} \][/tex]
3. Simplify further, grouping like factors:
[tex]\[ = \frac{x(x - 2y)}{(2x + y)y} \][/tex]
### Step 4: Write the final simplified expression.
Bringing all these simplifications together, the final expression becomes:
[tex]\[ \frac{x(x^2 - 3xy + 2y^2)}{y(x^3 + x^2 y - xy - y^2)} \][/tex]
Hence the solution to the given mathematical expression is:
[tex]\[ \frac{x(x^2 - 3xy + 2y^2)}{y(x^3 + x^2 y - xy - y^2)} \][/tex]
This represents the detailed step-by-step simplification process for the given expression.
### Step 1: Simplify each fraction separately.
1. Simplify [tex]\(\frac{x^2 - 4y^2}{x + y}\)[/tex]
Notice that [tex]\(x^2 - 4y^2\)[/tex] is a difference of squares:
[tex]\[ x^2 - 4y^2 = (x - 2y)(x + 2y) \][/tex]
Therefore,
[tex]\[ \frac{x^2 - 4y^2}{x + y} = \frac{(x - 2y)(x + 2y)}{x + y} \][/tex]
This simplifies to:
[tex]\[ x - 2y \quad \text{if} \quad x + y \neq 0 \][/tex]
2. Simplify [tex]\(\frac{x^2 - xy}{2x^2 + 5xy + 2y^2}\)[/tex]
The expression [tex]\(x^2 - xy\)[/tex] can be factored as:
[tex]\[ x^2 - xy = x(x - y) \][/tex]
We factor the denominator [tex]\(2x^2 + 5xy + 2y^2\)[/tex]. This can be recognized as a quadratic trinomial which factors as:
[tex]\[ 2x^2 + 5xy + 2y^2 = (2x + y)(x + 2y) \][/tex]
Therefore,
[tex]\[ \frac{x^2 - xy}{2x^2 + 5xy + 2y^2} = \frac{x(x - y)}{(2x + y)(x + 2y)} \][/tex]
3. Simplify [tex]\(\frac{x^2 y - y^2}{y + 2x}\)[/tex]
Factor the numerator [tex]\(x^2 y - y^2\)[/tex]:
[tex]\[ x^2 y - y^2 = y(x^2 - y) = y(x + y)(x - y) \][/tex]
Thus,
[tex]\[ \frac{x^2 y - y^2}{y + 2x} = \frac{y(x + y)(x - y)}{y + 2x} \][/tex]
### Step 2: Perform the division and multiplication.
Since division by a fraction is equivalent to multiplication by its reciprocal, we rewrite the operation:
[tex]\[ \frac{x^2 - 4y^2}{x + y} \times \frac{x^2 - xy}{2x^2 + 5xy + 2y^2} \div \frac{x^2 y - y^2}{y + 2x} = \frac{x^2 - 4y^2}{x + y} \times \frac{x^2 - xy}{2x^2 + 5xy + 2y^2} \times \frac{y + 2x}{x^2 y - y^2} \][/tex]
Substitute the simplified forms we obtained:
[tex]\[ (x - 2y) \times \frac{x(x - y)}{(2x + y)(x + 2y)} \times \frac{y + 2x}{y(x - y)} \][/tex]
### Step 3: Simplify the resulting expression.
1. Combine similar terms and cancel:
[tex]\[ = (x - 2y) \times \frac{x(x - y)}{(2x + y)(x + 2y)} \times \frac{y + 2x}{y(x - y)} \][/tex]
2. Notice that [tex]\( (x - y) \)[/tex] appears in both numerator and denominator, so they cancel:
[tex]\[ = (x - 2y) \times \frac{x}{(2x + y)(x + 2y) \times y} \][/tex]
3. Simplify further, grouping like factors:
[tex]\[ = \frac{x(x - 2y)}{(2x + y)y} \][/tex]
### Step 4: Write the final simplified expression.
Bringing all these simplifications together, the final expression becomes:
[tex]\[ \frac{x(x^2 - 3xy + 2y^2)}{y(x^3 + x^2 y - xy - y^2)} \][/tex]
Hence the solution to the given mathematical expression is:
[tex]\[ \frac{x(x^2 - 3xy + 2y^2)}{y(x^3 + x^2 y - xy - y^2)} \][/tex]
This represents the detailed step-by-step simplification process for the given expression.
Thank you for participating in our discussion. We value every contribution. Keep sharing knowledge and helping others find the answers they need. Let's create a dynamic and informative learning environment together. IDNLearn.com is your reliable source for accurate answers. Thank you for visiting, and we hope to assist you again.
