Rational Algebraic Expression

Rational Algebraic Expressions

Created by SGPELARIS​

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​What is a rational algebraic expression?

A rational algebraic expression is a fraction containing polynomials in

both numerator and denominator, provided that the denominator must not be equal to zero.

In a rational algebraic expression written in the form of P/Q where P and 𝑄 are polynomials, the polynomial 𝑄 is equal to any number except 0.

(Q ≠ 0).​

​Here’s a useful checklist in identifying whether the expression is a rational algebraic expression:

​

• The expression must be in fraction form.

• The expression must have in its numerator and denominator a constant,

  a variable, or a combination of both, that are polynomial expressions.​

• The expression must not have a negative exponent, a radical sign or a

  fraction exponent in the variable/s in both numerator and denominator.

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All of the expressions here are rational algebraic expressions since these contain polynomial

expressions in both numerator and denominator,

respectively.

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All of the expressions here are not rational algebraic expressions since the

expressions contain irrational numbers (√𝑥 and √𝑦) and variables having negative

exponents (𝑥⁻² and 𝑧⁻³), which are not polynomials.

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Multiple Choice

Which of the following terms is described as "the ratio of two polynomial

expressions"?

• A. linear algebraic expression

• B. linear algebraic equation

• C. rational algebraic expression

• D. rational algebraic equation

Multiple Choice

Which of the following is considered a rational algebraic expression?

 

• A. $$\frac{3n}{\sqrt[]{n}}$$  

• B. $$\frac{3n+9}{n^3\ -\ 2}$$  

• C. $$\frac{2x^{-2}-7x}{n^2}$$  

• D. $$\frac{4x\ +\ 11}{n^{-2}\ -1}$$  

Multiple Choice

Which of the following is a rational algebraic expression?

• A. $$\frac{6}{x\ -1}$$  

• B. $$\frac{2\ -\ \sqrt[]{x}}{4\ -\ x}$$  

• C. $$\frac{4\ +\ \frac{3}{x}}{y\ -\ 2}$$  

• D. $$\frac{1\ -\ x}{1\ +\ \frac{1}{x}}$$  

​Undefined RAE

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In rational algebraic expressions, you need to pay attention to what values

of the variables that will make the denominator equal to 0.

​

These values are called excluded values.

In rational algebraic expressions, the denominator cannot be equal to zero because a fraction having a denominator of 0 is undefined or meaningless.

​

​Steps in Determining the Excluded Values of RAE

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(Study Tip: Just pay attention to the denominator of the expression to determine

the excluded values.)​

Step 1: Let the expression in the denominator be equal to 0.

Step 2: Solve the equation to determine the value/s of the variable.​

Step 1: Let the expression in the denominator be equal to 0.

𝑥 − 3 = 0​

Step 2: Solve the equation to determine the value/s of the variable.​

​Example

​Solving for Excluded Values

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Multiple Choice

In a rational algebraic expression written in the form of P/Q where P and 𝑄 are polynomials, the polynomial 𝑄 is equal to any number except ________.

• A. 0

• B. 1

• C. -1

• D. 2

Multiple Choice

What value of x will make the rational algebraic expression $$\frac{5x\ +\ 3}{x\ -\ 6}$$  undefined?

• A. 5

• B. -5

• C. 6

• D. -6

​Simplifying RAE

A rational expression is said to be in simplified form when its numerator

and denominator have no common factor other than 1.

​

Steps on Simplifying Rational Expression​

1. Factor completely the numerator and denominator.​

2. Separate and divide out common factor/s if there is/are any.​

3. Multiply the remaining factors.​

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​Examples

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Multiple Choice

When is a rational algebraic expression in lowest term?

• A.  If the numerator and denominator are both of degree one.

• B. If either the numerator or the denominator is factored completely.B. If

• C. If the numerator and denominator have no common factor other  than 1

• D. If the numerator and denominator have no common factor other than −1.

Multiple Choice

Which of the following is one of the steps in simplifying rational expressions?

• A. Add the common factors.

• B. Multiply the common factors.

• C. Divide out the common factors.

• D. Subtract out the common factors.

Multiple Choice

Which of the following is the simplest form of $$\frac{p^3+q^3}{p^2-q^2}$$  ?

• $$\frac{p^2-pq\ +\ q^2}{p+q}$$  

• $$\frac{p^2+pq+q^2}{p+q}$$  

• $$\frac{p+q}{p-q}$$  

• $$\frac{p^2-pq\ +q^2}{p-q}$$  

​Operation of Rational Algebraic Expressions

​A. Multiplication of RAE

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Steps in multiplying rational algebraic expressions:

1. Factor each numerator and denominator completely.

2. Multiply the numerators and denominators.

3. Simplify by dividing out common factors.

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​Division of RAE

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​Example:

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​Addition and Subtraction of RAE

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Multiple Choice

What is the product of $$\frac{x-3}{x^2+x-20}\cdot\frac{x^2-8x+16}{x-3}$$  ?

• A. $$\frac{x-4}{x-5}$$  

• B. $$\frac{x-4}{x+5}$$  

• C. $$\frac{x+4}{x-5}$$  

• D. $$\frac{x+4}{x+5}$$  

Multiple Choice

Below are the steps in finding the quotients of rational algebraic expressions. Which of the following is in correct order?

I. Divide out common factors.

II. Simplify the remaining factors.

III. Find the reciprocal of the divisor and proceed to multiplication.

IV. Determine the dividend and divisor of the given expression.

• A. I, II, III, IV  

• B. II, III, IV, I    

• C. IV, III, I, II

•    D. IV, I, II, III

Multiple Choice

Find the truth about similar rational algebraic expressions among the

following statements.

                               

• A. The expressions have prime numerators.

• B. The expressions have prime denominators.

• C. The expressions have the same denominators.

• D. The expressions have the same numerators.

Multiple Choice

Perform the indicated operation :   $$\frac{x-2}{2}\div\frac{x^2\cdot4}{3}$$  

• A. $$\frac{3}{2x+4}$$  

• B. $$\frac{4}{2x-4}$$  

• C. $$\frac{6}{2\left(x=2\right)}$$  

• D. $$\frac{8}{2x-2}$$  

Multiple Choice

What is the correct rational expression of the phrase “the quotient of 2x and 3y ?

• A. $$\frac{3y}{2x}$$  

• B. $$\left(2x\right)\left(3y\right)$$  

• C. 2x - 3y

• D. $$\frac{2x}{3y}$$  

Multiple Choice

Write the verbal phrase into rational algebraic expression: “the average speed in x² +7 kilometers traveled in x - 2 hours.

•   A. $$\left(x^2-7\right)\left(x-2\right)$$  

• B. $$\frac{x-2}{x^2-7}$$  

• C. $$\frac{x^2-7}{x-2}$$  

• D. $$\frac{1}{x^2-7}\cdot x-2$$  

Multiple Choice

Write as one fraction and simplify : $$\frac{x}{x-1}-\frac{2}{x+1}$$  

• A. $$\frac{x^2+x+2}{\left(x+1\right)\left(x1\right)}$$  

• B. $$\frac{x^2-x+2}{\left(x+1\right)\left(x-1\right)}$$  

• C. $$\frac{x^2-x-2}{\left(x+1\right)\left(x-1\right)}$$  

• D. $$\frac{x^2+x-2}{\left(x+1\right)\left(x-1\right)}$$