Simplify the expression

Question

How do you simplify the rational expression x^{2 +} 4xy + 4y² over x ² + 3xy + 2y²

Answer 1

So -- the problem as you've stated is as follows:

x² + 4xy + 4y²

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x² + 3xy + 2y²

Simplify this rational expression.

In a rational expression, both the numerator and the denominator contain polynomials!

If you are solving this problem, I would assume you understand how polynomials can be factored, but I will be a little more detailed anyway.

In the numerator, you have (x² + 4xy + 4y²) — you want to factor this polynomial. Often times we make educated guesses and test for the correct answer with the FOIL method. Here, I would guess this factoring:

(x + 2y)(x + 2y) as the number 4 informs us that *at least* one of the variables will need to have a coefficient larger than 1.

Using the FOIL method - (First times First, Outside times Outside, Inside times Inside, Last times Last) we can confirm that multiplying the factors suggested - (x + 2y)(x + 2y) - does result in x² + 2xy + 2xy + 4y² - which, when like are combined, results in x² + 4xy + 4y².

Thus we have (x + 2y)(x + 2y) in the numerator:

(x + 2y)(x + 2y)

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x² + 3xy + 2y²

Using the same method to find the factors for the polynomial in the denominator, we can separate the polynomial into factors (x + y) and (x + 2y). When using the FOIL method, we find x² + xy + 2xy + 2y² which is further simplified into x² + 3xy + 2y² - thus, we know the factors for the denominator!

(x + 2y)(x + 2y)

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(x + y)(x + 2y)

When you have a factor being multiplied in the numerator AND the denominator, they can be cancelled out! Think of it this way - any number that has the same number in the numerator and the denominator can be considered one, and thus they "cancel each other out".

Your result as such:

(x + 2y)(x + 2y)

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(x + y)(x + 2y)

(x + 2y)

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(x + y)

Definitely let me know if I can clarify anything for you!

Answer 2

Answer 3

Write x^{2 +} 4xy + 4y²

as (x+2y) the whole squared

Rewrite x ² + 3xy + 2y²

^{as a product of two factors - (x+y)(x+2y)}

Then cancel the common factors, you'll be left with (x+2y)/(x+y)

Hope this helps