## How to Simplify Rational Expressions - Step-by-Step

In this section, we'll break down the process of simplifying rational expressions into clear and manageable steps. Follow along to learn how to simplify rational expressions step-by-step.

Remember, by taking the time to understand the explanation, you’re not just memorizing formulas but gaining a deeper understanding of the underlying concepts.

### Example 1

**Equation**: *x*2−9*x*2−6*x*+9*x*2−9

1. Factor both the numerator and the denominator.

**Numerator**: x2−9 factors into (x−3)(x+3)**Denominator**: x2−6x+9 factors into (x−3)2

2. List any restricted values to avoid division by zero.

- Since the denominator has a factor of x−3,x≠3x=3 to avoid division by zero.

3. Cancel common factors.

- Cancel one factor of x−3 from both the numerator and the denominator.

**4. Final Simplified Expression**: x+3x-3 with x=3.

**5. Explanation**: This simplification shows that by canceling common factors, we reduce the expression to its lowest terms, ensuring to note any restrictions to the domain of the expression.

**Equation**: *x*2−4*x*2−5*x*+6

1. Factor both the numerator and the denominator.

**Numerator**: *x*2−4 factors into (*x*−2)(*x*+2)**Denominator**: *x*2−5*x*+6 factors into (*x*−2)(*x*−3)

2. List any restricted values to avoid division by zero.

- Since the denominator has factors of x−2 and x-3, x ≠ 2 to avoid division by zero.

3. Cancel common factors.

- Cancel one factor of x−2 from both the numerator and the denominator.

**4. Final Simplified Expression**: x+2x-3 with x=2 and x=3.

**5. Explanation**: This simplification reduces the expression to its most basic form while clearly identifying any values of x that are not allowed to ensure the expression remains valid.

**Equation**: 3x2+12x6x2−9x

1. Factor both the numerator and the denominator.

- Numerator: 3x2+12x can be factored out to 3x(x+4)
- Denominator: 6x2−9x can be factored out to 3x(2x−3)

2. List any restricted values to avoid division by zero.

- Since the denominator includes a factor of x and 2x−3, x≠0 and x=2 32 to prevent division by zero.

3. Cancel common factors.

- Cancel the common factor of 3x from both the numerator and the denominator.

**4. Final Simplified Expression**: x+42x-3 with x≠0 and x≠32.

**Explanation**: By eliminating the common factors, we are left with a simpler expression that clearly specifies conditions under which the variable

x must not be valued to maintain a valid expression.

### Example 2

**Equation**: 3*x*2−27*x*2−9

Factor both the numerator and the denominator.

**Numerator**: 3x2−27 can be factored out as 3(x2−9).**Denominator**: x2−9 factors into (x−3)(x+3).

1. List any restricted values to avoid division by zero.

- Since the denominator factors into x−3 and x+3, x≠3 and x≠-3 to avoid division by zero.

2. Cancel common factors.

- The factor x2−9 is present in both the numerator and the denominator and can be canceled out.

3. **Final Simplified Expression**: 3 with x=3 and x=−3.

4. **Explanation**: This process highlights the importance of factoring and canceling identical factors in both the numerator and the denominator while also being mindful of the values that are excluded from the domain of the expression.

**Equation**: 2*x*2−8*x*2−4*x*+4

1. Factor both the numerator and the denominator.

**Numerator**: 2x2−8 can be factored out as 2(x2−4), which further factors into 2(x−2)(x+2).**Denominator**: x2−4x+4 factors into (x−2)2.

2. List any restricted values to avoid division by zero.

- Since the denominator factors into (x−2)2, x=2to avoid division by zero.

3. Cancel common factors.

- The factor x-2 is present in both the numerator and the denominator and can be canceled out.

**4. Final Simplified Expression**: 2(x+2)x-2 with x≠2.

**Explanation**: This example demonstrates the method of simplifying expressions by factoring and then canceling out identical factors from the numerator and the denominator. It emphasizes the necessity of identifying and excluding values that make the denominator zero, to ensure the expression remains defined.

**Equation**: 4*x*2−162*x*2−8*x*+8

1. Factor both the numerator and the denominator.

**Numerator**: 4x2−16 can be factored out as 4(x2−4), which further factors into 4(x−2)(x+2). **Denominator**: 2x2−8x+8 can be factored out as 2(x2−4x+4) which factors into 2(x−2)2.

2. List any restricted values to avoid division by zero.

- Since the denominator factors into 2(x−2)2,x≠2x=2 to avoid division by zero.

3. Cancel common factors.

- The factor x−2 is present in both the numerator and the denominator and can be canceled out, noting that one instance of x−2 remains in the denominator due to its squared presence.

**4. Final Simplified Expression**: 4(x+2)2(x-2) with x≠2

**Explanation**: Through this process, we see the importance of fully factoring the numerator and the denominator to find and cancel common factors. This not only simplifies the expression to its more fundamental form but also highlights the critical step of identifying values that must be excluded to maintain a valid and defined mathematical expression.

**Equation**: 5x2−20x2−4

1. Factor both the numerator and the denominator.

- Numerator: 5x2−20 can be factored out as 5(x2−4) which further factors into 5(x−2)(x+2).
- Denominator: x2−4 factors into (x−2)(x+2).

2. List any restricted values to avoid division by zero.

- Since the denominator factors into (x−2)(x+2),x≠2 and x≠−2 to avoid division by zero.

3. Cancel common factors.

- The factor x-2 and x+ 2 are present in both the numerator and the denominator and can be canceled out.

**4. Final Simplified Expression**: 5 with x=2 and x=−2.

**5. Explanation**: This example illustrates the process of factoring and canceling out common factors to simplify a rational expression, while also highlighting the importance of identifying restricted values to maintain the validity of the expression.

**Equation**: 6x2−24x2x2−8

1. Factor both the numerator and the denominator.

**Numerator**: 6x2−24x can be factored out as 6x(x−4)**Denominator**: 2x2−8 factors into 2(x2−4) which further factors into 2(x−2)(x+2).

2. List any restricted values to avoid division by zero.

- Since the denominator factors into 2(x−2)(x+2), x≠2 and x=−2 to avoid division by zero.

3. Cancel common factors.

- The factor
*x*−2 and *x*+2 are present in both the numerator and the denominator and can be canceled out.

4. Final Simplified Expression:

**5. Explanation**: Through this example, we demonstrate the process of simplifying rational expressions by factoring and canceling common factors, while ensuring to identify and exclude values that would result in division by zero. This ensures the validity and definition of the expression.