Explore: Simplifying Rational Expressions with Multiplication and Division Solutions

Simplifying Rational Expressions with Multiplication and Division Solutions

A simplified rational expression has no common factors between the numerator and denominator other than 1 or $- 1 .$
To simplify a rational expression:

1. Factor the numerator and denominator completely.
2. Determine if there are any restrictions on the domain.
3. Divide common factors out of the expression.

Rational expressions can have common factors that are monomials, binomials, trinomials, etc.
To simplify monomials:
- Use the rules you have already learned to simplify fractions .

- In this example, the coefficients were simplified, and the exponents were subtracted.

$\frac{{}^{1}8 x^{9}}{{}_{3}24 x^{5}} = \frac{1 x^{9 - 5}}{3} = \frac{x^{4}}{3}$

To simplify polynomials:
- The numerator and denominator must have identical factors in order to divide them out from the rational expression.

- In this example, once factored, only the identical linear binomials can be simplified out of the expression.

$\frac{(x - 3) (x + 7)}{(2 x - 3) (x + 7)} = \frac{x - 3}{2 x - 3}$

- It may be helpful to classify expressions by degree and number of terms verbally to help determine if they are identical.

Example 3

Simplify. State the restrictions on the denominator.

$\frac{2 x (x + 3) (3 x + 11)}{8 x^{2} (2 x - 5) (x + 3)}$

Plan
Solve for the restrictions on the denominator

Simplify the rational expression

Implement

$\begin{matrix} 8 x^{2} = 0 & 2 x - 5 = 0 & x + 3 = 0 \\ x = 0 & x = \frac{5}{2} & x = - 3 \end{matrix} x \neq - 3, 0, \frac{5}{2}$

$\frac{2 x (x + 3) (3 x + 11)}{2 x (4 x) (2 x - 5) (x + 3)}$

$\frac{(x + 3) (3 x + 11)}{4 x (2 x - 5) (x + 3)}$

$\frac{3 x + 11}{4 x (2 x - 5)}$ ,

$\frac{3 x + 11}{4 x (2 x - 5)}, x \neq - 3, 0, \frac{5}{2}$

Explain

Solve for the x-values in the denominator to name the restrictions
Simplify the monomial $\frac{2 x}{8 x^{2}}$
Simplify identical binomials
Write the answer with the restrictions

Note

See if you can find the restrictions using mental math.

Simplifying Rational Expressions with Multiplication and Division (cont.) Solutions

When a division symbol is present in a rational expression: 
- Find the reciprocal of the fraction after the symbol (the divisor). 
- And change the operation to multiplication . 
Therefore, when rational expressions are divided , state the restrictions for the entire divisor . 
Which means, you must determine all the values that are excluded from the numerator and the denominator of the divisor .

Example 4

Simplify the expression. State the restrictions on the denominator.

$\frac{2 x + 10}{12 x - 28} \div \frac{x^{2} - 25}{6 x - 14}$

Implement

$\frac{2 (x + 5)}{4 (3 x - 7)} \div \frac{(x - 5) (x + 5)}{2 (3 x - 7)}$

divisor (numerator and denominator)

$3 x - 7 = 0$

$x = \frac{7}{3}$

$x - 5 = 0$

$x = 5$

$x + 5 = 0$

$x = - 5$

$3 x - 7 = 0$

$x = \frac{7}{3}$

$x \neq \pm 5, \frac{7}{3}$

$\frac{2 (x + 5)}{4 (3 x - 7)} \cdot \frac{2 (3 x - 7)}{(x - 5) (x + 5)}$

$\frac{4 (x + 5) (3 x - 7)}{4 (3 x - 7) (x - 5) (x + 5)}$

$\frac{1}{x - 5}, x \neq \pm 5, \frac{7}{3}$

Explain 

Factor all parts of the expression 
State the restrictions for first denominator and the entire divisor 
Write the fraction after the division symbol as its reciprocal. 
Write as one large expression
Simplify like terms and expressions 
Write answer with the restrictions

Note

Anytime an answer contains both the positive and negative value, use the ± symbol.

Example 5

Simplify. State when the given expression is undefined.

$\frac{x^{2} + 6 x - 16}{2 x^{2} + 5 x - 3} \div \frac{3 x^{2} - 5 x - 2}{4 x^{2} - 36} \cdot \frac{x^{3} + 3 x^{2} - 10 x}{6 x^{2} + 12 x - 90}$

Implement

$\frac{(x + 8) (x - 2)}{(2 x - 1) (x + 3)} \div \frac{(3 x + 1) (x - 2)}{4 (x^{2} - 9)} \cdot \frac{x (x^{2} + 3 x - 10)}{6 (x^{2} + 2 x - 15)}$

$\frac{(x + 8) (x - 2)}{(2 x - 1) (x + 3)} \div \frac{(3 x + 1) (x - 2)}{4 (x - 3) (x + 3)} \cdot \frac{x (x + 5) (x - 2)}{6 (x + 5) (x - 3)}$

$\begin{matrix} 2 x - 1 = 0 & x + 3 = 0 & x - 3 = 0 & x + 3 = 0 \\ x = \frac{1}{2} & x = - 3 & x = 3 & x = - 3 \\ x + 5 = 0 & x - 3 = 0 & 3 x + 1 = 0 & x - 2 = 0 \\ x = - 5 & x = 3 & x = - \frac{1}{3} & x = 2 \end{matrix}$

$\frac{(x + 8) (x - 2)}{(2 x - 1) (x + 3)} \cdot \frac{4 (x - 3) (x + 3)}{(3 x + 1) (x - 2)} \cdot \frac{x (x + 5) (x - 2)}{6 (x + 5) (x - 3)}$

$\frac{2 (2 x) (x + 8) (x - 2) (x - 3) (x + 3) (x + 5) (x - 2)}{2 (3) (2 x - 1) (x + 3) (3 x + 1) (x - 2) (x + 5) (x - 3)} \frac{2 x (x + 8) (x - 2)}{3 (2 x - 1) (3 x + 1)}, x \neq - 5, - 3, - \frac{1}{3}, \frac{1}{2}, 2, 3$

Explain 

Factor all parts of the expression 
Factor completely 
State the restrictions

Note

Repeated values for the restrictions only need to be listed one time.

First and last denominator, middle expression: numerator and denominator
Write the fraction after the division symbol as its reciprocal
Write as one large expression
Simplify like terms and expressions

Note

This step is optional but may help you see that once the problem is written using multiplication, any identical term/expression in the numerator can be simplified.

Note

See Lesson 3 More to Explore for how to check your solutions using technology.