Subido por Odette Alegato

chapter 2

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Eng’g 10
PRE-CALCULUS
Rational Expressions
Pre-Calculus
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Learning Outcomes
After careful study of this chapter,
students should be able to do the follow
ing:
1. Perform calculations on Radicals by
applying the properties;
2. Solve problems on complex numbers.
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A. Rational Expressions
A rational expression is an
expression
that is the ratio of two
polynomials.
Example:
It is just like a fraction, but with
polynomials.
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A. Rational Expressions
In General
A rational function is the ratio of two
polynomials P(x) and Q(x) like this
𝑁(𝑥)
𝑓 𝑥 =
𝐷(𝑥)
Except that 𝐷(𝑥) cannot be zero since
1
= undefined.
0
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A. Rational Expressions
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A. Rational Expressions
Simplification of Rational Expressions
To simplify rational expressions we
must follow the following steps:
1. Factor numerator and denominator.
2. Simplify.
Example 1: Simplify the following rational
expression (x² + 8x + 12) / (x² - 36).
We factor the numerator and the
denominator,
remembering
that
(a² - b²) = (a + b) (a - b):
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Simplification of Rational Expressions
2
𝑥 + 8𝑥 + 12
𝑥+6 𝑥+2
=
𝑥 2 − 36
𝑥−6 𝑥+6
Cancel the common factor 𝑥 + 6
𝑥 2 + 8𝑥 + 12
𝒙+𝟐
=
2
𝑥 − 36
𝒙−𝟔
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Simplification of Rational Expressions
Example 2: Simplify the following rational
expression (n³ - n) / (n² - 5n - 6).
We factor the numerator and the
denominator,
𝑛3 − 𝑛
𝑛(𝑛 + 1)(𝑛 − 1)
=
𝑛2 − 5𝑛 − 6
𝑛−6 𝑛−1
Cancel the common factor 𝑛 − 1
3
𝑛 −𝑛
𝒏(𝒏 + 𝟏)
=
𝑛2 − 5𝑛 − 6
𝒏−𝟔
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Operations on Rational Expressions
Addition
The easiest way to add rational
expressions is to use the common
denominator.
Example 1:
2
3
2 𝑥+1 +3 𝑥−2
+
=
𝑥−2 𝑥+1
𝑥−2 𝑥+1
Then simplify to:
2
3
2𝑥 + 2 + 3𝑥 − 6
5𝑥 − 4
+
=
= 2
𝑥−2 𝑥+1
𝑥−2 𝑥+1
𝑥 −𝑥−2
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Operations on Rational Expressions
Subtraction
Subtracting is just like Adding:
Example 2:
2
3
2 𝑥+1 −3 𝑥−2
−
=
𝑥−2 𝑥+1
𝑥−2 𝑥+1
Then simplify to:
2
3
2𝑥 + 2 − (3𝑥 − 6)
−𝑥 + 8
+
=
= 2
𝑥−2 𝑥+1
𝑥−2 𝑥+1
𝑥 −𝑥−2
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Operations on Rational Expressions
Multiplication
To multiply two rational expressions, just
multiply the tops and bottoms
separately
Example 3:
2
𝑥−2
3
2 3
=
𝑥+1
𝑥−2 𝑥+1
Then simplify to:
2
𝑥−2
Rational Expressions
3
6
= 2
𝑥+1
𝑥 −𝑥−2
Pre-Calculus
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Operations on Rational Expressions
Division
To divide two rational expressions, first
flip the second expression over (make it a
reciprocal) and then multiply.
Example 4:
2
3
2
/
=
𝑥−2
𝑥+1
𝑥−2
𝑥+1
3
Then simplify to:
2
3
2𝑥 + 2
/
=
𝑥−2
𝑥+1
3𝑥 − 6
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B. Radicals
A radical is an expression of
𝑛
𝑎 denoting the principal 𝑛𝑡ℎ root
positive integer 𝑛 is the index, or
the radical and the number 𝑎 is the
The index is omitted if 𝑛 = 2.
the form
of 𝑎. The
order, of
radicand.
The laws for radicals are obtained
directly from the laws for exponents by
means of the definition
𝑛
Rational Expressions
𝑎𝑚 =
𝑚
𝑎𝑛
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Properties of Radicals
If 𝑛 is even, assume a, b ≥ 0.
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Simplification of Radicals
It is important to reduce a radical
to its simplest form through use of the
following operations.
1. Removal of perfect 𝑛𝑡ℎ 𝑝𝑜𝑤𝑒𝑟𝑠 from a
radicand. Any radical of order n should
be simplified by removing all perfect 𝑛𝑡ℎ
powers from under the radical sign
using the rule
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Simplification of Radicals
Examples:
2. Reduction of the index of the radical.
Examples:
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Simplification of Radicals
A radical is said to be in simplest
form if:
1. All perfect n-th powers have been
removed from the radical.
2. The index of the radical is as small
as possible.
3. No fractions are present in the
radicand i.e. the denominator has
been rationalized.
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Simplification of Radicals
Similar Radicals
Radicals which, on being reduced
to simplest form, have the same index
and radicand.
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C. Operations on Radicals
Addition and Subtraction of Radicals
Before addition or subtraction of
radicals it is important to reduce them
to simplest form. Like radicals can then
be added or subtracted in the same
way as other like terms.
Example:
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Operations on Radicals
Multiplication of Radicals
1. To multiply two or more radicals
having
the
same
index
use
𝑛
𝑛
𝑛
𝑎 𝑏 = 𝑎𝑏
Example:
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Multiplication of Radicals
1. To multiply radicals with different
indices use fractional exponents
and the laws of exponents.
Example:
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Operations on Radicals
Division of Radicals
1. To divide two radicals having the
same index use
𝑛
𝑛
𝑎
𝑏
=
𝑛
𝑎
,
𝑏
and
simplify.
Example:
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Division of Radicals
2. To divide radicals with different
indices use fractional exponents
and the laws of exponents.
Example:
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Rationalization
In one form of fraction, the
denominator is a binomial in which one
term is a square
root of a rational
number or expression and the other
term is of the same form or is rational
i.e. the denominator has the form
𝑎 + 𝑏 or 𝑎 + 𝑏 . In such a case,
rationalize the fraction by multiplying
the numerator and denominator by the
conjugate of the denominator.
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Rationalization
where the conjugate of 𝑎 + 𝑏 = 𝑎 −
𝑏 and the conjugate of
𝑎+𝑏 = 𝑎−
𝑏.Example:
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D. Complex Numbers
A Complex Number is a combination
of a Real Number and an Imaginary
Number.
Either Part Can Be Zero
So, a Complex Number has a real
part and an imaginary part.
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Complex Numbers
But either part can be 0, so all
Real Numbers and Imaginary Numbers
are also Complex Numbers.
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Complex Numbers
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Complex Numbers
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Complex Numbers
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Complex Numbers
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Complex Numbers
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Complex Numbers
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Complex Numbers
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Complex Numbers
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Complex Numbers
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Complex Numbers
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Complex Numbers
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Complex Numbers
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References
• Feliciano and Uy, College Algebra
• Dugopolski, Mark. College Algebra,
3rd ed. Addison-Wesley, 2002.
• Mijares, Catalina. College Algebra.
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For queries contact
Engr. ODETTE R. ALEGATO
email add: oralegato@mmsu.edu.ph
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