Dividing Rational Expressions: Step-by-Step Solutions

Hey guys! Today, we're diving deep into the world of dividing rational expressions. If you've ever felt a bit lost when faced with these problems, don't worry! We're going to break it down step by step, so you'll be solving them like a pro in no time. We will solve three different expressions, showing all steps.

Expression 1: $\frac{y^2 - 9}{y^2 + y} : \frac{y^2 + 6y + 9}{3y + 3} = ?$

Let's start with our first expression. Remember, dividing by a fraction is the same as multiplying by its reciprocal. This is a crucial concept, so keep it in mind as we move forward.

Step 1: Rewrite as Multiplication

The first thing we need to do is rewrite the division as multiplication by inverting the second fraction. This gives us:

$\frac{y^2 - 9}{y^2 + y} * \frac{3y + 3}{y^2 + 6y + 9}$

Step 2: Factor Everything

Now, let's factor all the polynomials we can. Factoring is key to simplifying these expressions. Look for differences of squares, common factors, and trinomial squares.

• $y^2 - 9$ is a difference of squares, so it factors into $(y - 3)(y + 3)$.
• $y^2 + y$ has a common factor of $y$, so it factors into $y(y + 1)$.
• $3y + 3$ has a common factor of $3$, so it factors into $3(y + 1)$.
• $y^2 + 6y + 9$ is a perfect square trinomial, so it factors into $(y + 3)^2$ or $(y + 3)(y + 3)$.

Substituting these factored forms into our expression, we get:

$\frac{(y - 3)(y + 3)}{y(y + 1)} * \frac{3(y + 1)}{(y + 3)(y + 3)}$

Step 3: Simplify by Cancelling Common Factors

Next, we simplify the expression by canceling out common factors in the numerator and denominator. This is where the magic happens, guys! We can cancel out:

• $(y + 3)$ from the numerator and denominator.
• $(y + 1)$ from the numerator and denominator.

This leaves us with:

$\frac{(y - 3)}{y} * \frac{3}{(y + 3)}$

Step 4: Multiply Remaining Factors

Finally, multiply the remaining factors in the numerator and the denominator:

$\frac{3(y - 3)}{y(y + 3)}$

So, the simplified form of the first expression is $\frac{3(y - 3)}{y(y + 3)}$. Remember, it's all about factoring and canceling, guys!

Expression 2: $\frac{xy+y^2}{a-3b} : \frac{x^2 - y^2}{2a - 6b} = ?$

Alright, let's move on to our second expression. We'll use the same steps as before, so you'll get plenty of practice.

Step 1: Rewrite as Multiplication

First, rewrite the division as multiplication by the reciprocal:

$\frac{xy + y^2}{a - 3b} * \frac{2a - 6b}{x^2 - y^2}$

Step 2: Factor Everything

Now, let's factor those polynomials:

• $xy + y^2$ has a common factor of $y$, so it factors into $y(x + y)$.
• $a - 3b$ is already in its simplest form.
• $2a - 6b$ has a common factor of $2$, so it factors into $2(a - 3b)$.
• $x^2 - y^2$ is a difference of squares, so it factors into $(x - y)(x + y)$.

Substitute the factored forms into our expression:

$\frac{y(x + y)}{a - 3b} * \frac{2(a - 3b)}{(x - y)(x + y)}$

Step 3: Simplify by Cancelling Common Factors

Now for the fun part – canceling! We can cancel out:

• $(x + y)$ from the numerator and denominator.
• $(a - 3b)$ from the numerator and denominator.

This leaves us with:

$\frac{y}{1} * \frac{2}{(x - y)}$

Step 4: Multiply Remaining Factors

Multiply the remaining factors:

$\frac{2y}{x - y}$

So, the simplified form of the second expression is $\frac{2y}{x - y}$. See how factoring makes everything so much easier, guys?

Expression 3: $\frac{a - 4b}{x^2 - 2xy + y^2} : \frac{4ab - a^2}{x - y} = ?$

Last but not least, let's tackle our third expression. By now, you should be getting the hang of this!

Step 1: Rewrite as Multiplication

Rewrite the division as multiplication:

$\frac{a - 4b}{x^2 - 2xy + y^2} * \frac{x - y}{4ab - a^2}$

Step 2: Factor Everything

Let's factor those polynomials:

• $a - 4b$ is already in its simplest form.
• $x^2 - 2xy + y^2$ is a perfect square trinomial, so it factors into $(x - y)^2$ or $(x - y)(x - y)$.
• $x - y$ is already in its simplest form.
• $4ab - a^2$ has a common factor of $a$, so it factors into $a(4b - a)$. Notice that $4b-a$ is the negative of $a-4b$, this will be useful for simplification

Substitute the factored forms:

$\frac{a - 4b}{(x - y)(x - y)} * \frac{x - y}{a(4b - a)}$

Step 3: Simplify by Cancelling Common Factors

Now, let's cancel those common factors. Remember, $a-4b$ and $4b-a$ are negatives of each other, so dividing them results in $-1$:

• $(a - 4b)$ and $(4b - a)$ cancel to $-1$.
• One $(x - y)$ cancels from the numerator and denominator.

This leaves us with:

$\frac{1}{(x-y)} * \frac{1}{-a}$

Step 4: Multiply Remaining Factors

Multiply the remaining factors:

$\frac{-1}{a(x - y)}$

So, the simplified form of the third expression is $\frac{-1}{a(x - y)}$. Awesome job, guys! You've now seen how to divide rational expressions by mastering the art of factoring and canceling.

Key Takeaways

• Always rewrite division as multiplication by the reciprocal.
• Factor all polynomials completely.
• Cancel common factors in the numerator and denominator.
• Multiply the remaining factors.

Dividing rational expressions might seem daunting at first, but with practice, you'll become a pro. Keep these steps in mind, and you'll be solving these problems with confidence. Remember, practice makes perfect, guys! So, keep at it, and you'll master this skill in no time.

If you have any questions or want to dive deeper into algebra, feel free to ask! Let's keep learning and growing together. You've got this!