Multiplying Rational Expressions Worksheet (pdf) with Answer Key ... - Free Printable

You've uploaded a math problem with two multiplication tasks involving rational expressions (fractions with polynomials). Let’s solve each one step by step, factoring and simplifying where possible.

---

Problem 18:

\[
\frac{3x^2 + 15x}{x^2 + x - 12} \cdot \frac{x^5 + 5x^2 + 4x}{x^2 + 4x - 5x}
\]

Step 1: Factor all numerators and denominators.

#### First fraction:

Numerator: \(3x^2 + 15x = 3x(x + 5)\)

Denominator: \(x^2 + x - 12\)

Factor: Look for two numbers that multiply to -12 and add to +1 → 4 and -3

So: \(x^2 + x - 12 = (x + 4)(x - 3)\)

#### Second fraction:

Numerator: \(x^5 + 5x^2 + 4x\)

Factor out the GCF: \(x(x^4 + 5x + 4)\)

Wait — let’s check that again. The expression is:

\(x^5 + 5x^2 + 4x\)

Actually, it's better to factor out \(x\) first:

= \(x(x^4 + 5x + 4)\) — but this doesn’t look easily factorable. Wait — perhaps there’s a typo?

Let me re-read: “x⁵ + 5x² + 4x”

That seems odd — usually we’d expect something like \(x^5 + 5x^3 + 4x\) or similar for factoring nicely. But let’s proceed as written.

Alternatively, maybe it was meant to be \(x^3 + 5x^2 + 4x\)? That would make more sense.

But since you wrote “x⁵”, I’ll assume it’s correct.

So numerator: \(x(x^4 + 5x + 4)\) — not easily factorable over integers.

Denominator: \(x^2 + 4x - 5x = x^2 - x\)

Ah! Here’s an error in the original problem — it says:

> \(x^2 + 4x - 5x\)

That simplifies to: \(x^2 - x = x(x - 1)\)

So second fraction becomes:

\[
\frac{x(x^4 + 5x + 4)}{x(x - 1)}
\]

Now, putting both fractions together:

\[
\frac{3x(x + 5)}{(x + 4)(x - 3)} \cdot \frac{x(x^4 + 5x + 4)}{x(x - 1)}
\]

Cancel common factors:

- One \(x\) cancels top and bottom.

Result:

\[
\frac{3x(x + 5) \cdot (x^4 + 5x + 4)}{(x + 4)(x - 3)(x - 1)}
\]

But \(x^4 + 5x + 4\) does not factor nicely over integers. So unless there’s a typo, this is as simplified as it gets.

✔ However, I suspect a typo in the problem.

It’s very likely that the numerator was meant to be:

> \(x^3 + 5x^2 + 4x\)

Because that factors nicely!

Let’s assume that’s the case (since otherwise the problem is messy and unusual).

So if numerator is \(x^3 + 5x^2 + 4x\):

Factor: \(x(x^2 + 5x + 4) = x(x + 1)(x + 4)\)

Then second fraction becomes:

\[
\frac{x(x + 1)(x + 4)}{x(x - 1)} = \frac{(x + 1)(x + 4)}{(x - 1)} \quad \text{(after canceling } x\text{)}
\]

Now multiply with first fraction:

\[
\frac{3x(x + 5)}{(x + 4)(x - 3)} \cdot \frac{(x + 1)(x + 4)}{(x - 1)}
\]

Cancel \((x + 4)\):

\[
\frac{3x(x + 5)(x + 1)}{(x - 3)(x - 1)}
\]

✔ Final Answer (assuming typo corrected):

\[
\boxed{\frac{3x(x + 1)(x + 5)}{(x - 3)(x - 1)}}
\]

*(Note: If the original \(x^5\) is intentional, then the answer remains messy — but given context, typo assumption is reasonable.)*

---

Problem 20:

\[
\frac{x^2 - y^2}{x + 1} \cdot \frac{5xy^2 + y^2}{2x^2 + 2xy}
\]

Step 1: Factor everything.

#### First fraction:

Numerator: \(x^2 - y^2 = (x - y)(x + y)\) — difference of squares

Denominator: \(x + 1\) — already factored

#### Second fraction:

Numerator: \(5xy^2 + y^2 = y^2(5x + 1)\)

Denominator: \(2x^2 + 2xy = 2x(x + y)\)

So now the expression is:

\[
\frac{(x - y)(x + y)}{x + 1} \cdot \frac{y^2(5x + 1)}{2x(x + y)}
\]

Step 2: Cancel common factors.

We see \((x + y)\) appears in numerator and denominator → cancel them.

Result:

\[
\frac{(x - y) \cdot y^2(5x + 1)}{(x + 1) \cdot 2x}
\]

No other common factors.

✔ Final Answer:

\[
\boxed{\frac{y^2(x - y)(5x + 1)}{2x(x + 1)}}
\]

---

✔ Final Answers:

18. Assuming typo correction (\(x^5\) → \(x^3\)):

\[
\boxed{\frac{3x(x + 1)(x + 5)}{(x - 3)(x - 1)}}
\]

20.

\[
\boxed{\frac{y^2(x - y)(5x + 1)}{2x(x + 1)}}
\]

Let me know if you want to explore the version with \(x^5\) — though it’s unlikely intended.