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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.
3
𝑥+1
𝑥 2 − 4𝑥 − 21
𝑥2 − 9
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.
𝑥+3
𝑥 2 − 5𝑥
What values can x not be?
P5 – Rational Expressions
What value of x must be excluded from the domain of?
𝑥+2
𝑥+1
P5 – Rational Expressions
Properties of Rational Expressions:
The following rules only work if Q and S do NOT equal 0.
Equality:
𝑃
𝑄
=
𝑅
𝑆
Equivalent Expressions
𝑃
𝑄
=
𝑃𝑅
,𝑅
𝑄𝑅
Sign
𝑃
−
𝑄
=
𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑃𝑆 = 𝑄𝑅
−𝑃
𝑄
≠0
=
𝑃
−𝑄
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 – Simplify a Rational Expression
Simplify:
3𝑥 2 − 20𝑥 − 7
2𝑥 2 − 11𝑥 − 21
P5 – Operations on Rational Expressions
Operations on Rational Expressions
The following rules only work if Q and S do NOT equal 0.
Addition
𝑃
𝑄
+
𝑅
𝑄
𝑅
𝑄
𝑃+𝑅
𝑄
=
Subtraction
𝑃
𝑄
−
Multiplication
𝑃
𝑄
∙ =
𝑃∙𝑅
𝑄∙𝑆
Division
𝑃
𝑄
÷ =
𝑅
𝑆
𝑃
𝑄
𝑅
𝑆
=
𝑃−𝑅
𝑄
𝑆
𝑅
∙ =
𝑃𝑆
𝑄𝑅
Where 𝑅 ≠ 0
P5 – Operations on Rational Expressions
Multiply:
𝑥2 − 4
𝑥 2 − 11𝑥 + 28
∙ 2
2
𝑥 + 2𝑥 − 8 𝑥 − 5𝑥 − 14
P5 – Operations on Rational Expressions
Divide:
𝑥 2 + 6𝑥 + 9 𝑥 2 + 7𝑥 + 12
∙ 3
3
𝑥 + 27
𝑥 − 3𝑥 2 + 9𝑥
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.
5𝑥 𝑥
5𝑥 + 𝑥 6𝑥 𝑥
+
=
=
=
18 18
18
18 3
If the rational expressions do not have a common denominator
find the LCD:
1. Factor each denominator completely and express repeated
factors using exponential notation.
2. 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 Expressions
Add:
2𝑥 + 1 𝑥 + 2
+
𝑥−3 𝑥+5
P5 – Operations on Rational Expressions
Subtract:
39𝑥 + 36
23𝑥 − 16
− 2
2
𝑥 − 3𝑥 − 10 𝑥 − 7𝑥 + 10
P5 – Operations on Rational Expressions
Use the Order of Operations:
𝑥 + 3 𝑥 + 4 𝑥 2 + 5𝑥 + 4
−
÷ 2
𝑥 − 2 𝑥 − 1 𝑥 + 4𝑥 − 5
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…
1. Multiply by 1 in the form of the LCD.
1.
2.
Determine the LCD of all fractions in the complex fraction.
Multiply both the numerator and the denominator of the complex
fraction by the LCD.
2. Multiply the numerator by the reciprocal of the denominator.
1. Simplify the numerator to a single fraction and the denominator by a
single fraction.
2. Using the definition for dividing fractions, multiply the numerator by the
reciprocal of the denominator.
3. If possible, simplify the resulting rational expressions.
P5 – Complex Fractions
Simplify:
2
1
+
𝑥−2 𝑥
3𝑥
2
−
𝑥−5 𝑥−5
P5 – Complex Fractions
Simplify:
2𝑥
4−
𝑥−2
2−
𝑥
P5 – Complex Fractions
Simplify:
𝑐 −1
𝑎−1 + 𝑏 −1
P5 – Complex Fractions
The average speed for a round trip is given by the complex
fraction:
2
1
1
+
𝑣1 𝑣2
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
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 of i.
If a is a positive real number, then −𝑎 = 𝑖 𝑎 The number 𝑖 𝑎 is
called an imaginary number.
−36 = 𝑖 36 = 6𝑖
−18 = 𝑖 18 = 3𝑖 2
P6 – Complex Numbers
A complex number is a number of the form a-bi, where a and b
are real numbers and 𝑖 = −1. The number a is the real part of
the a + bi, and b is the imaginary part.
- 3 + 5i
Real Part _____; Imaginary Part______
2 - 6i
Real Part _____; Imaginary Part______
5
Real Part _____; Imaginary Part______
7i
Real Part _____; Imaginary Part______
P6 – Complex Numbers
Writing a complex number in standard form a – bi.
7 + −45
P6 – Complex Numbers
Addition and Subtraction of Complex Numbers:
Basically add/subtract the real number parts and the
imaginary number parts.
7 − 2𝑖 + (−2 + 4𝑖)
P6 – Complex Numbers
−9 + 4𝑖 − (2 − 6𝑖)
P6 – Complex Numbers
Multiply
−6 ∙ −24
P6 – Complex Numbers
Multiply Complex Numbers…
Memorize this… 𝑖 2 = −1
3𝑖(2 − 5𝑖)
P6 – Complex
Numbers
2
Memorize this… 𝑖 = −1
3 − 4𝑖 2 + 5𝑖
P6 – Complex3 Numbers
Recall that the number
2
is not in simplest form because there
3
𝑖
is a radical expression in the denominator. Similarly is not in
simplest form because 𝑖 = −1.
3 𝑖 3𝑖
3𝑖
∙ = 2=
= −3𝑖
𝑖 𝑖 𝑖
−1
P6 – Complex Numbers
Simplify
3 − 6𝑖
2𝑖
P6 – Complex Numbers
Recall to simplify this;
2+ 3
,
5+ 3
we would multiply the numerator and
denominator by the conjugate of 5 + 3, which is 5 − 3.
What happens when we multiply a complex number by its
conjugate?
2 + 5𝑖 2 − 5𝑖
P6 – Complex Numbers
Divide the Complex Numbers
16 − 11𝑖
5 + 2𝑖
P6 – Complex Numbers
Powers of i:
𝑖1 = 1
𝑖 2 = −1
𝑖 3 = −𝑖
𝑖4 = 1
𝑖5 = 𝑖
𝑖 6 = −1
𝑖 7 = −𝑖
𝑖8 = 1
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.
𝑖153
So 153÷4 = 38 remainder 1; therefore, 𝑖1 = 1
Homework
• Continue finding news articles for quarter project.
• Chapter P Review Exercises
• 103 – 120 ALL