MTH120 College Algebra Chapter 1.6

 1.6 Rational Expressions:

Simplifying rational expressions involves reducing fractions that have polynomials in both the numerator and the denominator. The goal is to express the rational expression in its simplest form, which means factoring both the numerator and denominator, canceling out common factors, and eliminating any unnecessary terms. Here's a step-by-step guide on how to simplify rational expressions:

Step 1: Factor the Numerator and Denominator:

• Factor both the numerator and denominator completely into their prime factors. This may involve using techniques like factoring by grouping, difference of squares, or sum/difference of cubes.

Step 2: Cancel Common Factors:

• Identify any common factors between the numerator and denominator. Cancel out these common factors. This is similar to reducing fractions.
• Be sure to only cancel factors that are identical in both the numerator and denominator.

Step 3: State Any Restrictions:

• Identify any values of the variable(s) in the original rational expression that would make the denominator equal to zero. These values are excluded from the domain of the simplified expression because division by zero is undefined.

Step 4: Simplify the Expression:

• Write the simplified rational expression by removing any canceled factors and simplifying the remaining expression as much as possible.

Step 5: State the Domain:

• State the domain of the simplified expression, which is the set of all real numbers except for the values that were restricted in Step 3.

Let's go through an example to illustrate these steps:

Example: Simplify the rational expression $\frac{2{�}^2-4�}{4{�}^2-16}$.

Step 1: Factor the Numerator and Denominator:

• Factor the numerator: $2�(�-2)$.
• Factor the denominator: $4({�}^2-4)=4(�-2)(�+2)$.

Step 2: Cancel Common Factors:

• Cancel the common factor of $(�-2)$ in both the numerator and denominator.

Step 3: State Any Restrictions:

• The denominator cannot be zero, so we have $4(�-2)(�+2)\mathrm{\ne }0$. This implies $�\mathrm{\ne }2$ and $�\mathrm{\ne }-2$.

Step 4: Simplify the Expression:

• After canceling the common factor, we are left with $\frac{2�}{4}$.

Step 5: State the Domain:

• The domain of the simplified expression is all real numbers except $�=2$ and $�=-2$.

So, the simplified rational expression is $\frac{�}{2}$, and the domain is $\{�\in \mathbb{�}\mathrm{\mid }�\mathrm{\ne }2,�\mathrm{\ne }-2\}$≠−2}.

Let's go through a few examples of simplifying rational expressions:

Example 1: Simplify the rational expression $\frac{3{�}^2-12�}{6{�}^2+9�-15}$.

Step 1: Factor the Numerator and Denominator:

• Factor the numerator: $3{�}^2-12�=3�(�-4)$.
• Factor the denominator: $6{�}^2+9�-15=3(2{�}^2+3�-5)$. Further factoring the quadratic inside the parentheses: $2{�}^2+3�-5=(2�-1)(�+5)$.

Step 2: Cancel Common Factors:

• Cancel the common factor of 3 in both the numerator and denominator.

Step 3: State Any Restrictions:

• The denominator cannot be zero. Solve for $�$ in $2�-1=0$: $�\mathrm{\ne }\frac{1}{2}$.

Step 4: Write the Simplified Expression:

• After canceling the common factor, the simplified expression is $\frac{�-4}{2{�}^2+3�-5}$.

Step 5: State the Domain:

• The domain of the simplified expression is all real numbers except $�=\frac{1}{2}$, which was excluded in Step 3. So, the domain is $\{�\in \mathbb{�}\mathrm{\mid }�\mathrm{\ne }\frac{1}{2}\}$.

Example 2: Simplify the rational expression $\frac{5{�}^3-15{�}^2}{10{�}^2-25�}$.

Step 1: Factor the Numerator and Denominator:

• Factor the numerator: $5{�}^3-15{�}^2=5{�}^2(�-3)$.
• Factor the denominator: $10{�}^2-25�=5�(2�-5)$.

Step 2: Cancel Common Factors:

• Cancel the common factor of $5�$ in both the numerator and denominator.

Step 3: State Any Restrictions:

• The denominator cannot be zero. Solve for $�$ in $5�=0$: $�\mathrm{\ne }0$.

Step 4: Write the Simplified Expression:

• After canceling the common factor, the simplified expression is $\frac{�-3}{2�-5}$.

Step 5: State the Domain:

• The domain of the simplified expression is all real numbers except $�=0$, which was excluded in Step 3. So, the domain is $\{�\in \mathbb{�}\mathrm{\mid }�\mathrm{\ne }0\}$

Example 3: Simplify the rational expression $\frac{4{�}^2-16}{12{�}^2+8�}$.

Step 1: Factor the Numerator and Denominator:

• Factor the numerator: $4{�}^2-16=4({�}^2-4)=4(�-2)(�+2)$.
• Factor the denominator: $12{�}^2+8�=4�(3�+2)$.

Step 2: Cancel Common Factors:

• Cancel the common factor of 4 in both the numerator and denominator.

Step 3: State Any Restrictions:

• The denominator cannot be zero. Solve for $�$ in $4�=0$: $�\mathrm{\ne }0$.

Step 4: Write the Simplified Expression:

• After canceling the common factor, the simplified expression is $\frac{�-2}{3�+2}$.

Step 5: State the Domain:

• The domain of the simplified expression is all real numbers except $�=0$, which was excluded in Step 3. So, the domain is $\{�\in \mathbb{�}\mathrm{\mid }�\mathrm{\ne }0\}$.

These examples demonstrate how to simplify rational expressions by factoring, canceling common factors, and stating the domain restrictions.

Multiplying rational expressions involves multiplying two fractions where both the numerator and the denominator are polynomials. To multiply rational expressions, follow these steps:

Step 1: Factor the Numerators and Denominators:

• Factor each of the numerators and denominators completely into their prime factors. Use techniques like factoring by grouping, difference of squares, or other factorization methods.

Step 2: Cancel Common Factors:

• Identify and cancel any common factors between numerators and denominators across the expressions you're multiplying. Cancel only the factors that are identical in both the numerators and denominators.

Step 3: Multiply the Remaining Factors:

• Multiply the remaining factors in the numerators to get the new numerator, and multiply the remaining factors in the denominators to get the new denominator.

Step 4: Simplify the Result:

• If possible, simplify the resulting expression by canceling any common factors in the numerator and denominator.

Step 5: State Any Restrictions:

• Identify any values of the variable(s) that would make the denominator equal to zero in the original expressions. These values should be excluded from the domain of the simplified expression because division by zero is undefined.

Let's work through an example to illustrate these steps:

Example: Multiply the rational expressions $\frac{3{�}^2-9}{{�}^2-4}$ and $\frac{{�}^2+2�}{{�}^2-�-6}$.

Step 1: Factor the Numerators and Denominators:

• For the first expression:

  • Factor the numerator: $3{�}^2-9=3({�}^2-3)$.
  • Factor the denominator: ${�}^2-4=(�-2)(�+2)$.

• For the second expression:

  • Factor the numerator: ${�}^2+2�=�(�+2)$.
  • Factor the denominator: ${�}^2-�-6=(�-3)(�+2)$.

Step 2: Cancel Common Factors:

• In the first expression, cancel the common factor of 3 in the numerator and the common factor of $(�+2)$ in the denominator.
• In the second expression, cancel the common factor of $�(�+2)$ in the numerator and the common factor of $(�+2)$ in the denominator.

Step 3: Multiply the Remaining Factors:

• Multiply the remaining factors:
  • Numerator: $1\cdot 1=1$.
  • Denominator: $(�-2)(�-3)$.

Step 4: Simplify the Result:

• The result is $1$ over $(�-2)(�-3)$.

Step 5: State Any Restrictions:

• In the original expressions, the denominators cannot be zero. Solve for $�$ in $(�-2)(�+2)=0$ and $(�-3)(�+2)=0$:

  • $�\mathrm{\ne }2$ and $�\mathrm{\ne }-2$
  • 
    
  
  ≠−2 for the first expression.
  • $�\mathrm{\ne }3$ and $�\mathrm{\ne }-2$
  • 
    

• So, the domain of the simplified expression is {x∈R∣x≠2,x≠3,x≠−2}.$\{�\in \mathbb{�}\mathrm{\mid }�\mathrm{\ne }2,�\mathrm{\ne }3,�\mathrm{\ne }-2\}$

  
  

• 
  
• 
  
• 
  

The simplified result is $\frac{1}{(�-2)(�-3)}$ with the given domain restrictions.

Dividing rational expressions involves dividing one fraction (rational expression) by another. To divide rational expressions, follow these steps:

Step 1: Flip the Second Rational Expression:

• To divide one rational expression by another, first keep the first expression as it is.

Step 2: Change Division to Multiplication:

• Instead of dividing, change the division operation to multiplication by multiplying by the reciprocal (flipped) of the second rational expression.

Step 3: Factor the Numerators and Denominators:

• Factor each of the numerators and denominators in both rational expressions completely into their prime factors. Use factoring techniques like factoring by grouping, difference of squares, or other factorization methods.

Step 4: Cancel Common Factors:

• Identify and cancel any common factors between numerators and denominators across the expressions you're multiplying. Cancel only the factors that are identical in both the numerators and denominators.

Step 5: Multiply the Remaining Factors:

• Multiply the remaining factors in the numerators to get the new numerator, and multiply the remaining factors in the denominators to get the new denominator.

Step 6: Simplify the Result:

• If possible, simplify the resulting expression by canceling any common factors in the numerator and denominator.

Step 7: State Any Restrictions:

• Identify any values of the variable(s) that would make the denominator equal to zero in the original expressions. These values should be excluded from the domain of the simplified expression because division by zero is undefined.

Let's work through an example to illustrate these steps:

Example: Divide the rational expressions $\frac{3{�}^2-9}{{�}^2-4}$ by $\frac{{�}^2+2�}{{�}^2-�-6}$.

Step 1: Flip the Second Rational Expression:

• Keep the first expression as it is: $\frac{3{�}^2-9}{{�}^2-4}$.

Step 2: Change Division to Multiplication:

• Instead of dividing, change the operation to multiplication by multiplying by the reciprocal (flipped) of the second expression: $\frac{3{�}^2-9}{{�}^2-4}\cdot \frac{{�}^2-�-6}{{�}^2+2�}$.

Step 3: Factor the Numerators and Denominators:

• Factor each of the numerators and denominators:
  • First expression:
    • Numerator: $3({�}^2-3)$.
    • Denominator: $(�-2)(�+2)$.
  • Second expression:
    • Numerator: $�(�-3)$.
    • Denominator: $�(�+2)$.

Step 4: Cancel Common Factors:

• In the numerators, cancel the common factor of 3 and the common factor of $�$.
• In the denominators, cancel the common factor of $�$ and the common factor of $�+2$.

Step 5: Multiply the Remaining Factors:

• Multiply the remaining factors in the numerators and denominators: $\frac{(�-3)}{(�-2)}$

Step 6: Simplify the Result:

• The simplified result is $\frac{(�-3)}{(�-2)}$.

Step 7: State Any Restrictions:

• In the original expressions, the denominator cannot be zero. Solve for $�$ in $(�-2)(�+2)=0$:

  • $�\mathrm{\ne }2$ and $�\mathrm{\ne }-2$≠2.

• So, the domain of the simplified expression is $\{�\in \mathbb{�}\mathrm{\mid }�\mathrm{\ne }2,�\mathrm{\ne }-2\}${x∈R∣x≠2,x≠−2}.

The simplified result is $\frac{�-3}{�-2}$ with the given domain restrictions.

Let's work through a couple of examples of dividing rational expressions with step-by-step solutions.

Example 1: Divide the rational expressions $\frac{3{�}^2-12�}{2{�}^2+5�-3}$ by $\frac{{�}^2-4}{{�}^2+3�+2}$.

Step 1: Flip the Second Rational Expression:

• Keep the first expression as it is: $\frac{3{�}^2-12�}{2{�}^2+5�-3}$.

Step 2: Change Division to Multiplication:

• Change the operation to multiplication by multiplying by the reciprocal (flipped) of the second expression: $\frac{3{�}^2-12�}{2{�}^2+5�-3}\cdot \frac{{�}^2+3�+2}{{�}^2-4}$.

Step 3: Factor the Numerators and Denominators:

• Factor each of the numerators and denominators:
  • First expression:
    • Numerator: $3�(�-4)$.
    • Denominator: $(2�-1)(�+3)$.
  • Second expression:
    • Numerator: $(�+2)(�+1)$.
    • Denominator: $(�+2)(�-2)$.

Step 4: Cancel Common Factors:

• In the numerators, cancel the common factor of $�$ and the common factor of $�+2$.
• In the denominators, cancel the common factor of $�+2$.

Step 5: Multiply the Remaining Factors:

• Multiply the remaining factors in the numerators and denominators: $\frac{3(�-4)(�+1)}{(2�-1)(�-2)}$

Step 6: Simplify the Result:

• The simplified result is $\frac{3(�-4)(�+1)}{(2�-1)(�-2)}$.

Step 7: State Any Restrictions:

• In the original expressions, the denominators cannot be zero. Solve for $�$ in $(2�-1)(�-2)=0$:

  • $�\mathrm{\ne }\frac{1}{2}$ and $�\mathrm{\ne }2$.

• So, the domain of the simplified expression is $\{�\in \mathbb{�}\mathrm{\mid }�\mathrm{\ne }\frac{1}{2},�\mathrm{\ne }2\}$.

Example 2: Divide the rational expressions $\frac{{�}^2-9}{{�}^2-4�+3}$ by $\frac{{�}^2-5�+6}{{�}^2-3�+2}$.

Step 1: Flip the Second Rational Expression:

• Keep the first expression as it is: $\frac{{�}^2-9}{{�}^2-4�+3}$.

Step 2: Change Division to Multiplication:

• Change the operation to multiplication by multiplying by the reciprocal (flipped) of the second expression: $\frac{{�}^2-9}{{�}^2-4�+3}\cdot \frac{{�}^2-3�+2}{{�}^2-5�+6}$.

Step 3: Factor the Numerators and Denominators:

• Factor each of the numerators and denominators:
  • First expression:
    • Numerator: $(�-3)(�+3)$.
    • Denominator: $(�-1)(�-3)$.
  • Second expression:
    • Numerator: $(�-2)(�-1)$.
    • Denominator: $(�-3)(�-2)$.

Step 4: Cancel Common Factors:

• In the numerators, cancel the common factor of $(�-3)$ and the common factor of $(�-2)$.
• In the denominators, cancel the common factor of $(�-3)$ and the common factor of $(�-1)$.

Step 5: Multiply the Remaining Factors:

• Multiply the remaining factors in the numerators and denominators: $\frac{(�+3)}{(�-1)}$

Step 6: Simplify the Result:

• The simplified result is $\frac{(�+3)}{(�-1)}$.

Step 7: State Any Restrictions:

• In the original expressions, the denominators cannot be zero. Solve for $�$ in $(�-1)(�-3)=0$:

  • $�\mathrm{\ne }1$ and $�\mathrm{\ne }3$.

• So, the domain of the simplified expression is $\{�\in \mathbb{�}\mathrm{\mid }�\mathrm{\ne }1,�\mathrm{\ne }3\}$.

These examples demonstrate how to divide rational expressions with step-by-step solutions and domain restrictions.

Adding and subtracting rational expressions involves combining fractions where both the numerators and denominators are polynomials. To add or subtract rational expressions, follow these steps:

Step 1: Find a Common Denominator:

• Identify a common denominator for the expressions you are adding or subtracting. The common denominator should be a polynomial that both denominators can be expressed as a multiple of.

Step 2: Rewrite with a Common Denominator:

• Rewrite each rational expression so that it has the common denominator. To do this, multiply both the numerator and the denominator of each expression by the necessary factor to make the denominators equal.

Step 3: Perform the Operation:

• After you have a common denominator, you can add or subtract the numerators while keeping the common denominator.

Step 4: Simplify the Result:

• If possible, simplify the resulting expression by factoring the numerator and denominator and canceling common factors.

Step 5: State Any Restrictions:

• Identify any values of the variable(s) that would make the common denominator equal to zero. These values should be excluded from the domain of the simplified expression because division by zero is undefined.

Let's work through an example of both addition and subtraction of rational expressions:

Example 1 (Addition): Add the rational expressions $\frac{2�}{{�}^2-9}$ and $\frac{�}{{�}^2-�-6}$.

Step 1: Find a Common Denominator:

• Identify a common denominator. In this case, the common denominator is $(�-3)(�+3)(�-2)$, which combines the factors of both denominators.

Step 2: Rewrite with a Common Denominator:

• Rewrite each rational expression with the common denominator: $\frac{2�}{{�}^2-9}=\frac{2�}{(�-3)(�+3)}$ $\frac{�}{{�}^2-�-6}=\frac{�}{(�-3)(�+2)}$

Step 3: Perform the Operation:

• Add the numerators while keeping the common denominator: $\frac{2�}{(�-3)(�+3)}+\frac{�}{(�-3)(�+2)}$

Step 4: Simplify the Result:

• To add the fractions, find a common denominator, which is already obtained in this case. $\frac{2�}{(�-3)(�+3)}+\frac{�}{(�-3)(�+2)}=\frac{2�+�}{(�-3)(�+3)}=\frac{3�}{(�-3)(�+3)}$

Step 5: State Any Restrictions:

• In the original expressions, the denominators cannot be zero. Solve for $�$ in $(�-3)(�+3)=0$:

  • $�\mathrm{\ne }3$ and $�\mathrm{\ne }-3$.

• So, the domain of the simplified expression is $\{�\in \mathbb{�}\mathrm{\mid }�\mathrm{\ne }3,�\mathrm{\ne }-3\}$.

Example 2 (Subtraction): Subtract the rational expressions $\frac{3�}{{�}^2-4}$ and $\frac{�}{{�}^2-�-6}$.

Step 1: Find a Common Denominator:

• Identify a common denominator. In this case, the common denominator is $(�-2)(�+2)(�-3)$, which combines the factors of both denominators.

Step 2: Rewrite with a Common Denominator:

• Rewrite each rational expression with the common denominator: $\frac{3�}{{�}^2-4}=\frac{3�}{(�-2)(�+2)}$ $\frac{�}{{�}^2-�-6}=\frac{�}{(�-3)(�+2)}$

Step 3: Perform the Operation:

• Subtract the numerators while keeping the common denominator: $\frac{3�}{(�-2)(�+2)}-\frac{�}{(�-3)(�+2)}$

Step 4: Simplify the Result:

• To subtract the fractions, find a common denominator, which is already obtained in this case.
• 
  

Here are two more examples of adding and subtracting rational expressions:

Example 3 (Addition): Add the rational expressions $\frac{2}{{�}^2-5�+6}$ and $\frac{3}{{�}^2-4}$.

Step 1: Find a Common Denominator:

• Identify a common denominator. In this case, the common denominator is $(�-2)(�-3)(�+2)$, which combines the factors of both denominators.

Step 2: Rewrite with a Common Denominator:

• Rewrite each rational expression with the common denominator: $\frac{2}{{�}^2-5�+6}=\frac{2}{(�-2)(�-3)}$ $\frac{3}{{�}^2-4}=\frac{3}{(�-2)(�+2)}$

Step 3: Perform the Operation:

• Add the numerators while keeping the common denominator: $\frac{2}{(�-2)(�-3)}+\frac{3}{(�-2)(�+2)}$

Step 4: Simplify the Result:

• To add the fractions, find a common denominator, which is already obtained in this case. $\frac{2}{(�-2)(�-3)}+\frac{3}{(�-2)(�+2)}=\frac{2+3}{(�-2)(�-3)}+\frac{5}{(�-2)(�+2)}=\frac{5}{(�-2)(�-3)}+\frac{5}{(�-2)(�+2)}$

Step 5: State Any Restrictions:

• In the original expressions, the denominators cannot be zero. Solve for $�$ in $(�-2)(�-3)=0$:

  • $�\mathrm{\ne }2$ and $�\mathrm{\ne }3$.

• So, the domain of the simplified expression is $\{�\in \mathbb{�}\mathrm{\mid }�\mathrm{\ne }2,�\mathrm{\ne }3\}$.

Example 4 (Subtraction): Subtract the rational expressions $\frac{{�}^2+5�+6}{{�}^2-�-6}$ and $\frac{{�}^2-4}{{�}^2-9}$.

Step 1: Find a Common Denominator:

• Identify a common denominator. In this case, the common denominator is $(�-3)(�+3)(�-2)$, which combines the factors of both denominators.

Step 2: Rewrite with a Common Denominator:

• Rewrite each rational expression with the common denominator: $\frac{{�}^2+5�+6}{{�}^2-�-6}=\frac{(�+3)(�+2)}{(�-3)(�+3)(�-2)}$ $\frac{{�}^2-4}{{�}^2-9}=\frac{(�+2)(�-2)}{(�-3)(�+3)(�-2)}$

Step 3: Perform the Operation:

• Subtract the numerators while keeping the common denominator: $\frac{(�+3)(�+2)}{(�-3)(�+3)(�-2)}-\frac{(�+2)(�-2)}{(�-3)(�+3)(�-2)}$

Step 4: Simplify the Result:

• To subtract the fractions, find a common denominator, which is already obtained in this case. $\frac{(�+3)(�+2)}{(�-3)(�+3)(�-2)}-\frac{(�+2)(�-2)}{(�-3)(�+3)(�-2)}=\frac{(�+3)(�+2)-(�+2)(�-2)}{(�-3)(�+3)(�-2)}$

Step 5: State Any Restrictions:

• In the original expressions, the denominators cannot be zero. Solve for $�$ in ((x - 3)(x + 3)(x - 2) =
• 
  

Simplifying complex rational expressions involves simplifying fractions where both the numerators and denominators are themselves rational expressions (fractions). The process is similar to simplifying regular fractions but requires factoring and reducing the fractions within the numerators and denominators. Here's a step-by-step guide:

Step 1: Factor the Numerators and Denominators:

• Begin by factoring each of the numerators and denominators in the complex rational expression completely into their prime factors. Use factoring techniques like factoring by grouping, difference of squares, or other factorization methods as needed.

Step 2: Simplify the Fractions within Fractions:

• Examine each fraction within the complex expression separately, both in the numerator and denominator. Simplify each fraction as much as possible. Cancel any common factors between the numerators and denominators of these internal fractions.

Step 3: Rewrite the Expression:

• Rewrite the complex rational expression with the simplified fractions within fractions, maintaining the original structure.

Step 4: Simplify Further if Possible:

• If there are still common factors that can be canceled between the numerators and denominators of the outermost fractions, simplify further.

Step 5: State Any Restrictions:

• Identify any values of the variable(s) that would make the denominators equal to zero in the original expressions or the simplified expression. These values should be excluded from the domain of the expression because division by zero is undefined.

Let's work through an example to illustrate these steps:

Example: Simplify the complex rational expression $\frac{\frac{{�}^2-4}{�-2}}{\frac{{�}^2+3�-4}{{�}^2-5�+6}}$.

Step 1: Factor the Numerators and Denominators:

• Factor each of the numerators and denominators within the complex expression:
  • For the numerator of the inner fraction: ${�}^2-4=(�+2)(�-2)$
  • For the denominator of the inner fraction: ${�}^2-5�+6=(�-2)(�-3)$
  • For the numerator of the outer fraction: ${�}^2+3�-4=(�+4)(�-1)$
  • For the denominator of the outer fraction: $�-2$ (no further factorization needed)

Step 2: Simplify the Fractions within Fractions:

• Simplify the inner fractions: $\frac{{�}^2-4}{�-2}=\frac{(�+2)(�-2)}{�-2}$ (Notice that $�-2$ cancels out.) $\frac{{�}^2+3�-4}{{�}^2-5�+6}=\frac{(�+4)(�-1)}{(�-2)(�-3)}$

Step 3: Rewrite the Expression:

• Rewrite the complex expression with the simplified inner fractions: $\frac{(�+2)(�-2)}{�-2}÷\frac{(�+4)(�-1)}{(�-2)(�-3)}$

Step 4: Simplify Further if Possible:

• Cancel the common factor of $(�-2)$ in the numerator and denominator of the outer fractions: $\frac{\cancel{(�+2)}\cancel{(�-2)}}{\cancel{(�-2)}}÷\frac{(�+4)(�-1)}{\cancel{(�-2)}(�-3)}$

Step 5: State Any Restrictions:

• In the original expression, the denominators cannot be zero. Solve for $�$ in $�-2=0$ and $�-3=0$:

  • $�\mathrm{\ne }2$ and $�\mathrm{\ne }3$.

• So, the domain of the simplified expression is $\{�\in \mathbb{�}\mathrm{\mid }�\mathrm{\ne }2,�\mathrm{\ne }3\}$.

The simplified complex rational expression is $\frac{�+4}{�-1}$ with the given domain restrictions

Let's work through a couple more examples of simplifying complex rational expressions:

Example 2: Simplify the complex rational expression $\frac{\frac{3{�}^2-9}{{�}^2-4}}{\frac{{�}^2-5�+6}{{�}^2+2�}}$.

Step 1: Factor the Numerators and Denominators:

• Factor each of the numerators and denominators within the complex expression:
  • For the numerator of the inner fraction: $3{�}^2-9=3({�}^2-3)$
  • For the denominator of the inner fraction: ${�}^2-4=(�-2)(�+2)$
  • For the numerator of the outer fraction: ${�}^2-5�+6=(�-3)(�-2)$
  • For the denominator of the outer fraction: ${�}^2+2�=�(�+2)$

Step 2: Simplify the Fractions within Fractions:

• Simplify the inner fractions: $\frac{3({�}^2-3)}{(�-2)(�+2)}$ $\frac{(�-3)(�-2)}{�(�+2)}$

Step 3: Rewrite the Expression:

• Rewrite the complex expression with the simplified inner fractions: $\frac{3({�}^2-3)}{(�-2)(�+2)}÷\frac{(�-3)(�-2)}{�(�+2)}$

Step 4: Simplify Further if Possible:

• Cancel the common factors in the numerators and denominators: $\frac{3\cancel{({�}^2-3)}}{(�-2)\cancel{(�+2)}}÷\frac{\cancel{(�-3)}\cancel{(�-2)}}{�\cancel{(�+2)}}$

Step 5: State Any Restrictions:

• In the original expression, the denominators cannot be zero. Solve for $�$ in $�-2=0$ and $�+2=0$:

  • $�\mathrm{\ne }2$ and $�\mathrm{\ne }-2$.

• So, the domain of the simplified expression is $\{�\in \mathbb{�}\mathrm{\mid }�\mathrm{\ne }2,�\mathrm{\ne }-2\}$.

The simplified complex rational expression is $\frac{3�}{�}$ with the given domain restrictions. This simplifies further to $3$ because $�$ cancels out.

Example 3: Simplify the complex rational expression $\frac{\frac{{�}^2-1}{{�}^2-5�+6}}{\frac{{�}^2+�-6}{{�}^2-2�}}$.

Step 1: Factor the Numerators and Denominators:

• Factor each of the numerators and denominators within the complex expression:
  • For the numerator of the inner fraction: ${�}^2-1=(�-1)(�+1)$
  • For the denominator of the inner fraction: ${�}^2-5�+6=(�-2)(�-3)$
  • For the numerator of the outer fraction: ${�}^2+�-6=(�+3)(�-2)$
  • For the denominator of the outer fraction: ${�}^2-2�=�(�-2)$

Step 2: Simplify the Fractions within Fractions:

• Simplify the inner fractions: $\frac{(�-1)(�+1)}{(�-2)(�-3)}$ $\frac{(�+3)(�-2)}{�(�-2)}$

Step 3: Rewrite the Expression:

• Rewrite the complex expression with the simplified inner fractions: $\frac{(�-1)(�+1)}{(�-2)(�-3)}÷\frac{(�+3)(�-2)}{�(�-2)}$

Step 4: Simplify Further if Possible:

• Cancel the common factors in the numerators and denominators: $\frac{(�-1)(�+1)}{(�-2)\cancel{(�-3)}}÷\frac{\cancel{(�+3)}\cancel{(�-2)}}{�\cancel{(�-2)}}$

Step 5: State Any Restrictions:

• In the original expression, the denominators cannot be zero. Solve for $�$ in $�-2=0$ and $�-3=0$:

  • $�\mathrm{\ne }2$ and $�\mathrm{\ne }3$.

• So, the domain of the simplified expression is $\{�\in \mathbb{�}\mathrm{\mid }�\mathrm{\ne }2,�\mathrm{\ne }3\}$.

The simplified complex rational expression is $\frac{(�-1)(�+1)}{�}$ with the given domain restrictions

Algebra Unit 1 test review Question:(n/x)n Answer:x Question:n/x x n/y Answer:n/xy Question:n/y Answer:n/X Question:(x-h)2+(y-k)2 Answer:r2 Question:How do you find the slope? Answer: y2-y1 ------ = Slope x2-X1 Question:Where does an undefined slope cross? Answer:Crosses the x axis Question:What's the formula for point slope form? Answer:y=my+b Question:What's the formula for standard form? Answer:Ax+By=C Question:What's the formula for point slope form? Answer:Y-Y1=m(X-X1) Question:What's the formula for a horizontal line? Answer:Y=b Question:What's the formula for a vertical line? Answer:x=a Question:The x value tells Answer:left to right Question:The Y value tells Answer:Up And Down Question:What's the distance Formula? Answer:/(x2-X1)^2 + (y2-y1)^2 Question:What's The Midpoint Formula? Answer:X1+ X2/2, Y1 +Y2/2 Question:What's a problem in Inequality Notation? Answer:14<x<19 Question:What's a problem for interval notation? Answer:[14,19) Question:What's a closed interval? Answer:c<x<d Question:What's an open interval? Answer:S<x<t Question:What are Infinite intervals? Answer:[b,~) means >b,-