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College algebra . P.5 – Rational Expressions P.6 – Complex Numbers. P5 – Rational Expressions. A rational expression is a fraction in which the numerator and denominator are polynomials. P5 – Rational Expressions.

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College algebra

College algebra

P.5 – Rational Expressions

P.6 – Complex Numbers

P5 rational expressions
P5 – Rational Expressions

A rational expression is a fraction in which the numerator and denominator are polynomials.

P5 rational expressions1
P5 – Rational Expressions

The domain of a rational expression is the set of all real numbers that can be used as replacements for the variable. Any variable that causes division by zero is excluded from the domain of the rational expression.

What values can x not be?

P5 rational expressions2
P5 – Rational Expressions

What value of x must be excluded from the domain of?

P5 rational expressions3
P5 – Rational Expressions

Properties of Rational Expressions:

The following rules only work if Q and S do NOT equal 0.


Equivalent Expressions


P5 simplify a rational expression
P5 – Simplify a Rational Expression

To simplify a rational expression, factor the numerator and denominator. Then use the equivalent expressions property to eliminate factors common to both the numerator and denominator. A rational expression is simplified when 1 is the only common factor of both the numerator and the denominator.

P5 operations on rational expressions
P5 – Operations on Rational Expressions

Operations on Rational Expressions

The following rules only work if Q and S do NOT equal 0.




Division Where

P5 operations on rational expressions3
P5 – Operations on Rational Expressions

Addition of rational expressions with a common denominator is accomplished by writing the sum of the numerators over the common denominator.

If the rational expressions do not have a common denominator find the LCD:

  • Factor each denominator completely and express repeated factors using exponential notation.

  • Identify the largest power of each factor in any single factorization. The LCD is the product of each factor raised to its largest power.

P5 operations on rational expressions6
P5 – Operations on Rational Expressions

Use the Order of Operations:

P5 complex fractions
P5 – Complex Fractions

A complex fraction is a fraction whose numerator or denominator contains one or more fractions. Simplify complex fractions using one of the following…

  • Multiply by 1 in the form of the LCD.

    • Determine the LCD of all fractions in the complex fraction.

    • Multiply both the numerator and the denominator of the complex fraction by the LCD.

  • Multiply the numerator by the reciprocal of the denominator.

    • Simplify the numerator to a single fraction and the denominator by a single fraction.

    • Using the definition for dividing fractions, multiply the numerator by the reciprocal of the denominator.

    • If possible, simplify the resulting rational expressions.

P5 complex fractions4
P5 – Complex Fractions

The average speed for a round trip is given by the complex fraction:

where v1 is the average speed on the way to your destination and v2 is the average speed on your return trip. Find the average speed for a round trip of v1 = 50 mph and v2 = 40 mph.

P6 complex numbers
P6 – Complex Numbers

Definition of i

The imaginary unit, designated by the letter i is the number such that i2 = -1.

The principle square root of a negative number is defined in terms ofi.

If a is a positive real number, then The number is called an imaginary number.

P6 complex numbers1
P6 – Complex Numbers

A complex number is a number of the form a-bi, where a and b are real numbers and . The number a is the real part of the a + bi, and b is the imaginary part.

- 3 + 5i Real Part _____; Imaginary Part______

2 - 6iReal Part _____; Imaginary Part______

5Real Part _____; Imaginary Part______

7iReal Part _____; Imaginary Part______

P6 complex numbers2
P6 – Complex Numbers

Writing a complex number in standard form a – bi.

P6 complex numbers3
P6 – Complex Numbers

Addition and Subtraction of Complex Numbers:

Basically add/subtract the real number parts and the imaginary number parts.

P6 complex numbers6
P6 – Complex Numbers

Multiply Complex Numbers…

Memorize this…

P6 complex numbers7
P6 – Complex Numbers

Memorize this…

P6 complex numbers8
P6 – Complex Numbers

Recall that the number is not in simplest form because there is a radical expression in the denominator. Similarly is not in simplest form because

P6 complex numbers10
P6 – Complex Numbers

Recall to simplify this; , we would multiply the numerator and denominator by the conjugate of , which is .

What happens when we multiply a complex number by its conjugate?

P6 complex numbers11
P6 – Complex Numbers

Divide the Complex Numbers

P6 complex numbers12
P6 – Complex Numbers

Powers of i:

We can find all values of i to powers by dividing the power by 4. The remainder that is left will help us evaluate the value of i.

So 153÷4 = 38 remainder 1; therefore,


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