Download Absolute Value If a is a real number then the absolute value of a is |a

Absolute Value
If a is a real number then the absolute value of a is
(
a,
if a ≥ 0
|a| =
−a, if a < 0
• For every a ∈ R, |a| ≥ 0.
• For every a ∈ R, |a| = | − a|.
• For every a, b ∈ R, |ab| = |a||b|.
a |a|
.
• For every a ∈ R, =
b
|b|
Examples:
|0|
|3 − π|
|2 − | − 12||
| − x|
7 − 12 12 − 7 If a and b are in R, then the distance between points a and b on the real line
is
d(a, b) = |b − a|.
1
Exponents
If a ∈ R and n is a positive integer, then an = a
· · · · a}.
| · a{z
n times
How to handle negatives and inverses:
Let a, b ∈ R and n and m positive integers.
• (ab)n = an bn
a n an
•
= n
b
b
a −n b n
•
=
b
a
• a0 = 1
• a−n =
•
1
an
am
= am−n
an
• (am )n = amn
•
Examples:
Simplify the following expressions:
1
(2s3 t−2 )( s7 t)(16t4 )
4
x4 z 2
4y 5
2x3 y 2
z3
2
Eliminate the negative powers:
−1 −1 −2 q r s
r−5 sq −8
2
bm
a−n
=
b−m
an
Radicals
√
“
” means the positive square root of; that is,
√
a=b
means
b2 = a and b ≥ 0.
√
“ n ”, for a positive integer n reads “the principal n-th root of”, and it
means
√
n
a = b means bn = a and if n is even b ≥ 0.
How to handle radicals:
Let a, b ∈ R and n and m be positive integers.
√
√
√ √
n
n
• n an = a if n is odd
• ab = n a b
r
√
√
n
n
a
a
an = |a| if n is even
•
n
= √
•
n
b
b
q
√
n √
m
•
a = nm a
Examples: Simplify the following expressions:
p
x3 y 6
√
3
√
3
a2 b 64a4 b
3
Assume b ≥ 0, combine the radicals:
√
36b −
√
b3
Rational Exponents
For a positive integer n and a ∈ R
a1/n =
For any rational number
m
,
n
√
n
a.
n>0
am/n =
m √
√
n
a = n am
Examples: Simplify by writing the radicals as exponents
√
√
3
2( x)(5 x2 )
Examples: Rationalize the denominators.
1
√ , a≥0
a
2
√
3
x
4
Algebraic Expressions
A variable is a letter that can represent any number from a given set, for
example x or y.
An algebraic expression is a combination of variables and real numbers using
addition, subtraction, product, division and exponents.
√
x+
2
x
2y 4 + 5y + 3
x−3
x2 + 5
Polynomials
A polynomial in a variable x is an expression of the form
an xn + nn−1 xn−1 + · · · + a1 x + a0 ,
where ai ∈ R and n is a non-negative integer. If an 6= 0 the n is called the
degree of the polynomial.
(what is the difference between positive integers and non-negative integers?)
How to handle polynomials
We use the properties of real numbers, specially the distributive property, to
add, subtract and multiply algebraic expressions.
Examples: Expand each expression
(3x2 + x + 1) + (2x2 − 3x − 5)
8(2x + 5) − 7(x − 9)
(4x − 5y)(3x − y)
5
(2x + 3)(x2 − 5x + 4)
√
1
x3/2 ( x − √ )
x
In the above examples we are “expanding”. The reverse process is called
factoring.
Examples: Factor each expression
−2x3 + 16x
2x2 y − 6xy 2 + 3xy
To factor a polynomial of the form x2 + bx + c, we need to find two numbers
that add up to b and multiply to c.
x2 − 6x + 5
To factor a polynomial of the form ax2 + bx + c, we need to find factors
6
(px + r) and (qx + s) such that
ax2 + bx + c = (px + r)(qx + s) = pqx2 + (ps + qr)x + rs
That means we look for numbers p, q, r and s such that pq = a, ps + qr = b
and rs = c.
3x2 − 16x + 5
3(3y + 2)2 − 16(3y + 2) + 5
7