Simplifying Rational Expressions Worksheets - Math Monks - Free Printable

Let's solve each rational expression step by step and simplify them.

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1. $\frac{x^2 + 2x - 3}{x + 3}$

Step 1: Factor the numerator.

$$
x^2 + 2x - 3 = (x + 3)(x - 1)
$$

Step 2: Rewrite the expression:

$$
\frac{(x + 3)(x - 1)}{x + 3}
$$

Step 3: Cancel the common factor $(x + 3)$, assuming $x \neq -3$:

$$
x - 1
$$

✔ Answer: $\boxed{x - 1}$

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2. $\frac{x^2 - 16}{x - 4}$

Step 1: Recognize that $x^2 - 16$ is a difference of squares:

$$
x^2 - 16 = (x + 4)(x - 4)
$$

Step 2: Rewrite:

$$
\frac{(x + 4)(x - 4)}{x - 4}
$$

Step 3: Cancel $(x - 4)$, assuming $x \neq 4$:

$$
x + 4
$$

✔ Answer: $\boxed{x + 4}$

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3. $\frac{2x^2 + 7x + 6}{x^2 - x - 6}$

Step 1: Factor numerator and denominator.

Numerator: $2x^2 + 7x + 6$

Find two numbers that multiply to $2 \cdot 6 = 12$ and add to 7: $3$ and $4$

$$
2x^2 + 7x + 6 = 2x^2 + 4x + 3x + 6 = 2x(x + 2) + 3(x + 2) = (2x + 3)(x + 2)
$$

Denominator: $x^2 - x - 6 = (x - 3)(x + 2)$

Step 2: Rewrite:

$$
\frac{(2x + 3)(x + 2)}{(x - 3)(x + 2)}
$$

Step 3: Cancel $(x + 2)$, assuming $x \neq -2$:

$$
\frac{2x + 3}{x - 3}
$$

✔ Answer: $\boxed{\frac{2x + 3}{x - 3}}$

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4. $\frac{x^2 + 4x}{3x + 12}$

Step 1: Factor both numerator and denominator.

Numerator: $x^2 + 4x = x(x + 4)$

Denominator: $3x + 12 = 3(x + 4)$

Step 2: Rewrite:

$$
\frac{x(x + 4)}{3(x + 4)}
$$

Step 3: Cancel $(x + 4)$, assuming $x \neq -4$:

$$
\frac{x}{3}
$$

✔ Answer: $\boxed{\frac{x}{3}}$

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5. $\frac{y^2 + 6y}{yx + 6x}$

Step 1: Factor numerator and denominator.

Numerator: $y^2 + 6y = y(y + 6)$

Denominator: $yx + 6x = x(y + 6)$

Step 2: Rewrite:

$$
\frac{y(y + 6)}{x(y + 6)}
$$

Step 3: Cancel $(y + 6)$, assuming $y \neq -6$:

$$
\frac{y}{x}
$$

✔ Answer: $\boxed{\frac{y}{x}}$

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6. $\frac{x - 2}{2x^2 - 4x}$

Step 1: Factor denominator.

$$
2x^2 - 4x = 2x(x - 2)
$$

Step 2: Rewrite:

$$
\frac{x - 2}{2x(x - 2)}
$$

Step 3: Cancel $(x - 2)$, assuming $x \neq 2$:

$$
\frac{1}{2x}
$$

✔ Answer: $\boxed{\frac{1}{2x}}$

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7. $\frac{4x - 4}{6x - 20}$

Step 1: Factor numerator and denominator.

Numerator: $4x - 4 = 4(x - 1)$

Denominator: $6x - 20 = 2(3x - 10)$ → Wait, better to factor out GCF:

Actually: $6x - 20 = 2(3x - 10)$ — but let's check if we can factor further.

Wait: $6x - 20 = 2(3x - 10)$ — no common factors with numerator?

But let's see:

Numerator: $4(x - 1)$
Denominator: $2(3x - 10)$

No common binomial or variable terms.

Wait — maybe I made a mistake. Let's recheck:

Is there any common factor?

Numerator: $4x - 4 = 4(x - 1)$
Denominator: $6x - 20 = 2(3x - 10)$

No common binomial factor. But we can simplify constants:

$$
\frac{4(x - 1)}{2(3x - 10)} = \frac{2(x - 1)}{3x - 10}
$$

✔ Answer: $\boxed{\frac{2(x - 1)}{3x - 10}}$

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8. $\frac{x^2 - 3x - 40}{x^2 - 11x + 24}$

Step 1: Factor both numerator and denominator.

Numerator: $x^2 - 3x - 40$

Find two numbers that multiply to $-40$ and add to $-3$: $-8$ and $5$

So: $(x - 8)(x + 5)$

Denominator: $x^2 - 11x + 24$

Find two numbers that multiply to $24$ and add to $-11$: $-8$ and $-3$

So: $(x - 8)(x - 3)$

Step 2: Rewrite:

$$
\frac{(x - 8)(x + 5)}{(x - 8)(x - 3)}
$$

Step 3: Cancel $(x - 8)$, assuming $x \neq 8$:

$$
\frac{x + 5}{x - 3}
$$

✔ Answer: $\boxed{\frac{x + 5}{x - 3}}$

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✔ Final Answers:

1. $x - 1$
2. $x + 4$
3. $\frac{2x + 3}{x - 3}$
4. $\frac{x}{3}$
5. $\frac{y}{x}$
6. $\frac{1}{2x}$
7. $\frac{2(x - 1)}{3x - 10}$
8. $\frac{x + 5}{x - 3}$

> 🔔 Note: Always state restrictions where denominators were canceled (e.g., $x \neq -3$, $x \neq 4$, etc.), but since the problem only asks to simplify, these are implied.

Let me know if you'd like the restrictions listed too!