MHB What am I doing wrong? (Simplifying Rational Expressions)

AI Thread Summary
The discussion focuses on simplifying the rational expression and identifying mistakes in the process. The original expression involves two fractions that need to be simplified through factoring. Key errors include illegal cancellation of terms and incorrect multiplication of polynomials. The correct approach emphasizes factoring first to identify restrictions on the variables before simplification. Proper application of the FOIL method is also highlighted as essential for accurate multiplication of binomials.
eleventhxhour
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7a) Simplify and state any restrictions on the variables:

$$\frac{x^2-5xy+4y^2}{x^2+3xy-28y^2} ⋅ \frac{x^2+2xy+y^2}{x^2-y^2} $$

I'm not really sure what a good process would be to simplify this. This is what I tried to do, (below) which is wrong. Could anyone point out what I did wrong and what a better process to simplify it would be? Thanks!

My answer:

$$\frac{-5xy+4y^2}{3xy-28y^2} ⋅ \frac{2xy + y^2}{-y^2}$$

$$\frac{-10x^2y^2 + 4y^4}{-3xy^3+28y^4}$$

$$\frac{2y^2(-5x^2+2y^2}{y^3(-3x+28y)}$$

And then I'm not sure what to do.
 
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I would suggest factoring first, and then from this you can see the restrictions and then simplify:

$$\frac{(x-y)(x-4y)}{(x+7y)(x-4y)}\cdot\frac{(x+y)^2}{(x+y)(x-y)}$$
 
Hello, eleventhxhour!

What are you doing wrong? . . . Everything!

Simplify: .$$\frac{x^2-5xy+4y^2}{x^2+3xy-28y^2} ⋅ \frac{x^2+2xy+y^2}{x^2-y^2} $$
You have: .$$\frac{x^2-5xy+4y^2}{x^2+3xy-28y^2} ⋅ \frac{x^2+2xy+y^2}{x^2-y^2} $$

Then you canceled illegally:

. . $$\frac{{\color{red}\rlap{//}}x^2-5xy+4y^2}{{\color{red}\rlap{//}}x^2+3xy-28y^2} ⋅ \frac{{\color{red}\rlap{//}}x^2+2xy+y^2}{{\color{red}\rlap{//}}x^2-y^2} $$

And got: .[math]\frac{-5xy+4y^2}{+3xy-28y^2} ⋅ \frac{+2xy+y^2}{-y^2} [/math]Then you multiplied incorrectly.

You said: .[math](-5xy + 4y^2)(2xy + y^2) \:=\:-10x^2y + 4y^4[/math]

. . as if you never heard of "FOIL".
 
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