college algebra

Module 4: Discussion Solving rational equations

 Module 4:  Solving rational expressions, equations, and functions 

Discussion Instructions:

Please review  the class discussion directions . They tell you how to prepare and reflect before responding to the prompt below. They also include detailed guidelines how you must respond to classmates and then reply to the responses you receive. Please pay close attention to the respond directions. You must post, respond, and reply as directed to earn your discussion points .

Discussion writing prompt

1. Please solve this problem. Please show your work.

Revenue, cost, and profit are studied in economics. If R(x) is a revenue function and C(x) is a cost function, then the profit function, P(x) is equal to: P(x) = R(x) – C(x)

Where x is the number of items manufactured and sold by a company.

A company builds and sells six products each week; assume that:

R(x) = (6x-7)/(x+2)  and C(x) = (4x-13)/(x+3).  Determine the profit function.

___________________ Problem source: Angle, A., & Rundle, D. (2018).  Intermediate algebra for college students.

Response focus

1. In your response, in addition to the standard requirement of noting what you have in common and where you differ with classmates, offer a suggestion as to how to improve your classmates' comfort level with the topics we are studying this week.

Remember to review the academic expectations for your submission.

Submission reminders:

· Submit your initial discussion post by 11:59 pm ET on Wednesday.  Respond to two of your classmates' discussion posts, and reply to the responses you received by 11:59 pm ET on Sunday.

· Contribute a minimum of 250 words to the initial post.

· Follow established  netiquette (Links to an external site.)  guidelines when participating in forums.

Reply

Module 4:  Radical Expression Lecture

Rational Expression

Rational Expression is an expression of the form pq, where p and q are polynomials and q ≠ 0. (example: 2x )

Rational Function is a function of the form y=f(x)=pq, where p and q are polynomials and q ≠ 0 (example: f(x)=2x)

The domain of a rational function y=f(x)=pq is the set of all real numbers for which the denominator, q, is not equal to 0 (example: for the function f(x)=2x, the domain is {𝑥|𝑥 ≠0}

Rational Expressions Simplification

A rational expression is simplified when the numerator and denominator have no common factors other than 1

Example:

Simplify 69

69=23(in this example, 6 and 9 contain the common factor 3)

To simplify rational expressions:

1. Factor both numerators and denominators as completely as possible

2. Divide both the numerator and the denominator by any common factors

Another Example:

Simplify x2+5x+4x+4

Factor the numerator: (𝑥+4)(𝑥+1) (x+4)and divide by the common factor (𝑥+4) (x+4), the result is: (x+1)

 

Multiply Rational Expressions:

To multiply rational expressions we can use the following rule: ab⋅cd=a⋅cb⋅d, where b ≠ 0, d ≠ 0.

Steps

1. Factor all numerators and denominators as far as possible.

2. Divide out any common factors.

3. Multiply using the above rule.

4. Simplify the answer when possible.

Example:

Multiply: x−56x⋅x2−2xx2−7x+10

Factor: x−56x⋅x2−2xx2−7x+1−=x−56x⋅x(x−2)(x−2)(x−5)

Divide out common factors: x6x=16

Divide Rational Expressions

To multiply rational expressions we can use the following rule: ab÷cd=a⋅db⋅c, where b ≠ 0, c ≠ 0, d ≠ 0

To divide rational expressions, multiply the first rational expression by the reciprocal of the second rational expression.

Example:

Divide: 18x45y3÷3x525y

Steps:

1. Multiply by the reciprocal of the divisor: 18x45y3⋅25y3x5

2. Divide out common factors: 18x43x5⋅25y5y3=6x⋅5y2=30xy2

Addition and Subtraction of Rational Expressions with different denominators

Steps:

1. Determine the least common denominator (LCD).

2. Rewrite each fraction as an equivalent fraction with the LCD. This is done by multiplying both the numerator and denominator of each fraction by any factors needed to obtain the LCD.

3. Leave the denominator in factored form, but multiply out the numerator.

4. Add or subtract the numerators while maintaining the LCD.

5. When it is possible to reduce the fraction by factoring the numerator, do so.

Example

Add: 2x+9y

LCD = xy

1. write each fraction with the LCD by multiplying both numerator and denominator of each fraction by any factors needed to obtain the LCD.

2. Multiply the first fraction by yy and the second fraction by xx 2x+9y=yy⋅2x+9y⋅xx=2yxy+9xxy

3. Add the numerators while maintaining the LCD 2yxy+9xxy=2y+9xxyor9x+2yxy

Complex Fractions

A complex fraction is a fraction that has a rational expression in its numerator or its denominator, or both its numerator and denominator.

Examples: 254, x+1xx−2 , 8+1x+21x2

Simplify Complex Fractions by Multiplying by the LCD

Steps

1. Find the LCD of all fractions appearing within the complex fraction. This is the LCD of the complex fraction.

2. Multiply both the numerator and denominator of the complex fraction by the LCD of the complex fraction found in step 1

3. Simplify when possible

Example

Simplify: 4x2−3xx25

The denominators in the complex fraction are 𝑥2, x, and 5; therefore, the LCD is 5𝑥2

1. Multiply the numerator and denominator by 5𝑥2: 5x2(4x2−3x)5x2(x52)

2. Apply distributive property: 5x2(4x2)−5x2(3x)5x2(x25)

3. Simplify: 5(4)−5x(3)x4=20−15xx4

 

Division of Polynomials and Synthetic Division

In long division, follow these four steps:

1. Divide

2. Multiply

3. Subtract

4. Bring down

Example

Divide 2x3−8x2=9x−2x−2 using long division.

                   

Explanation Paragraph: x – 2 is called the divisor and 2 x3-8 x2  +9 x - 2 is called the dividend. The first step is to find what we need to multiply the first term of the divisor  (x) by to obtain the first term of the dividend (2 x3). This is 2 x2. We then multiply  x – 2 by 2 x2 and put this expression underneath the dividend. The term 2 x2 is part of the quotient, and is put on top of the horizontal line (above the 8 x2). We then  subtract 2x3 - 4 x2 from 2 x3 – 8 x2 + 9 x – 2.

 

The same procedure is continued until an expression of lower degree than the divisor is obtained. This is called the remainder.

We’ve found that 2x3−8x2+9x−2x−2=2x2−4x+1

Source: The Dolciani Mathematics Learning Center. (2022). Dividing polynomials using long division.  Hunter College.  http://www.hunter.cuny.edu/dolciani/pdf_files/brushup-materials/dividing-polynomials-using-long-division.pdfLinks to an external site.

Dividing polynomials using Synthetic Division

When a polynomial is divided by a binomial of the form x – a, the division process can be shortened by a process called synthetic division.

To see augment what is in your textbook on this topic, please visit this handout by Beth-Allyn Osikiewicz, Ph.D, Associate Professor of Mathematics at Kent State University, Tuscarawas:

Synthetic Division: The Remainder Theorem and the Factor Theorem (PDF, new tab)Links to an external site.

 

References

Angel, A., & Rundle, D. (2018).  Intermediate algebra for college students. (10th edition). Pearson.