Download Review of Algebra

FAKULTAS TEKNOLOGI INFORMASI
UNIVERSITAS KRISTEN SATYA WACANA
MATRIKULASI MATEMATIKA
Review of Algebra
Topic Review of Algebra
Arithmetic Operations
Fractions
Factoring
Completing The Square
Quadratic Formula
Binomial Theorem
Radicals
Exponents
Inequalities
Absolute Value
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Review of Algebra
Here we review the basic rules and procedures of algebra that
you need to know in order to be successful in calculus
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1. ARITHMETIC OPERATIONS
The real numbers have the following properties:
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In particular, putting a = -1 in Distributive Law, we get
and so
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Example 1
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If we use the Distributive Law three times, we get
This says that we multiply two factors by multiplying each term in
one factor by each term in the other factor and adding the
products.
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Schematically, we have
In this case where c = a and d = b, we have
or
(1)
Similarly, we obtain
(2)
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Example 2
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2. FRACTIONS
Two add two fraction with the same denominator, we use the
Distributive Law:
Thus, it is true that
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But remember to avoid the following common error:
(For instance, take a = b = c = 1 to see the error)
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To add two fraction with different denominators, we use a common
denominator:
We multiply such fractions s follows:
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In particular, it is true that
To divide two fractions, we invert and multiply:
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Example 3
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3. FACTORING
We have used the Distributive Law to expand certain algebraic
expressions. We sometimes need to reverse this process (again using
the Distributive Law) by factoring an expression as a product of simpler
ones.
The easiest situation occurs when the expression has a common factor
as follows:
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To factor a quadratic of the form x2 + bx + c we note that
so we need to choose numbers r and s so that r + s = b and rs = c.
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Example 4
Factor x2 + 5x - 24
Solution
The two integers that add to give 5 and multiply to give -24 are -3 and 8.
Therefore
x2 + 5x - 24 = (x - 3)(x + 8)
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Example 5
Factor 2x2 − 7x − 4
Solution
Even though the coefficient of x2 is not 1, we can still look for factors of the
from
2x + r and x + s, where rs = −4.
Experimentation reveals that
2x2 − 7x − 4 = (2x + 1)(x − 4)
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Some special quadratics can be factored by using Equations 1 or 2
(from right to left) or by using the formula for a difference of
squares:
(3)
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The analogous formula for a difference of cubes is
(4)
Which you can verify by expanding the right side. For a sum
of cubes we have
(5)
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Example 6
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Example 7
Simplify
Solution
Factoring numerator and denominator, we have
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The Factor Theorem
To factor polynomials of degree 3 or more, we sometimes use the
following fact.
If P is a polynomial and P(b) = 0, then x − b is a factor of P(x).
(6)
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Example 8
Factor x3 − 3x2 − 10x + 24
Solution
Let P(x) = x3 − 3x2 − 10x + 24.
If P(b) = 0, where b is an integer, then b is a factor of 24. Thus
possibilities for b are ∓1, ∓2, ∓3, ∓4, ∓6,∓8, ∓12,and ∓24.
We find P(1) = 12, P(-1) = 30, P(2) = 0.
By the Factor Theorem, (x − 2) is a factor.
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Instead of substituting further, we use long division as follows:
Therefore
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4. COMPLETING THE SQUARE
Completing the square is a useful technique for graphing parabolas or
integrating rational functions.
Completing the square means rewriting a quadratic ax2 + bx + c
in the form a(x + p)2 + q and can be accomplished by:
1.
2.
Factoring the number a from the terms involving x.
Adding and subtracting the square of half the coefficient of x.
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In general, we have
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Example 9
Rewrite x2 + x + 1 by completing the square.
Solution
The square of half the coefficient of x is 1/4. Thus
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Example 10
Rewrite 2x2 − 12x + 11 by completing the square.
Solution
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5. QUADRATIC FORMULA
By completing the square as above we can obtain the following
formula for the roots of a quadratic equation.
The Quadratic Formula
The roots of the quadratic equation ax2 + bx + c = 0 are
(7)
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Example 10
Solve the equation 5x2 + 3x − 3
Solution
With a = 5, b = 3, and c = -3, the quadratic formula gives the solutions
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The quantity b2 − 4ac that appears in the quadratic formula is called
the discriminant. There are three possibilities:
If b2 − 4ac > 0, the equation has two real roots.
If b2 − 4ac = 0, the roots are equal.
If b2 − 4ac < 0, the equation has no real root. (The roots are complex.)
In case (3) the quadratic ax2 + bx + c can’t be factored and is called
irreducible.
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These three cases correspond to the fact that the number of times the
parabola y = ax2 + bx + c crosses the x-axis is 2, 1, or 0 (see Figure 1).
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Example 12
The quadratic x2 + x + 2 is irreducible because its discriminant is
negative
Solution
Therefore, it is impossible to factor x2 + x + 2.
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