Download Pre-Calculus Review

Math 31
Pre-Calculus
Name: __________________
Pre-Calculus Review
Slope & Equations of Lines
Recall:
rise y y 2  y1


run x x2  x1

Slope formula: m 

Equation of a line:
o Slope-intercept form: y  mx  b (m is the ________; b is the ______________)
o Slope-point form: y  y1  m( x  x1 ) (m is the ________; x1 , y1  is a ________
____________________)
o General form: Ax  By  C  0
Class Example:
Write the equation of the line through the points (5, -6) and (-3, 0) in general form. What is its
slope?
1. State the slope of each linear relation:
a)
b) y  5 
f ( x)  2  3x
d) x  3
c) 4 x  6 y  5  0
2. Write the equation of each line in general form:
3
a) with a y-intercept of -6 and a slope of
5
1
3
( x  2)
4
Math 31
Pre-Calculus
b) through the point (-2, 4) and parallel to the line 2 x  4 y  8  0
c) through the points (-3, 5) and (4, -5)
(If you need more practice on this topic, do p. 9 #1-3)
Factoring
Recall:
 Factoring Methods:
- GCF
- trinomial factoring (PSF, grid)
- difference of squares
 The Factor Theorem: A polynomial P (x ) has x  b as a factor if and only if P(b)  0 .
Class Examples:
Factor the following completely:
a) 75  3x 2
b) 3n 2  17n  28
c) 2 x 3  13x 2  16 x  5
d) 3x 3  x 2  12 x  4
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Math 31
Pre-Calculus
NEW:
 Factoring a difference of cubes: a 3  b 3 

Factoring a sum of cubes: a 3  b 3 
Class Examples:
Factor the following:
a) x 3  27

b) 8 x 3 y  y 4
You can factor expressions that involve fractional or negative exponents by factoring out
a common factor with the lowest exponent.
Class Examples:
Factor the following completely:
3
1
a) x 2  10 x 2  25 x
 12
b) 2  128 x 3
3. Factor the following completely:
a) y 2  2 y  35
b) 49 x 2  9 y 2
c) n 4  16
d) 8 x 2  14 x  5
e) 49 x 1  x
f) 216  x3
g) 27 x 3  8
h) n 3  n 3  6n 3
7
3
4
1
Math 31
Pre-Calculus
i) 1  9 x 1  20 x 2
k)
x
2

1

 2 2  6 x2  2
j) x 4  5 x3  x  5

 12
l) 3x 3  10 x 2  x  12
(If you need more practice on this topic, do p. 3 #1-4)
Rational Expressions
Recall:
 Always factor before simplifying. You can never cancel half a binomial.

Adding and subtracting rational expressions requires a _____________
_____________________.

Non-permissible values are any that make the _____________________ equal zero.
Class Examples:
Simplify the following:
3n  9
a) 3
n  27
b)
4
2 x
1
x
x2
Math 31
Pre-Calculus
Perform the indicated operation. State the non-permissible values.
12 x
3x 4
x2
1
a) 2
b)


x 1 x 1
x4 x
4. Simplify the following:
x 2  3x
a)
x4
b)
1
5
x

1
c)
x3
d)
5
x 2  5x  6
x2  4
x 2  4x  3
1
1
x2
Math 31
Pre-Calculus
5. Perform the indicated operation. State the non-permissible values.
2 5y4
1
x2

a) 3 
b)
x  2 2x  3
8
y
c) x  4 
2x  1
x
d)
2
3

x  1 2x  2
2
Composition of Functions
Recall:
 In a composite function f g (x) or f  g , the function g (x ) replaces the variable in
f (x) .
Class Example:
For the following functions, find f  g , g  f , f  f , and g  g .
1
f ( x) 
, g ( x)  x
x4
6
Math 31
Pre-Calculus
6. For the following functions, find f  g , g  f , f  f , and g  g .
a) f ( x)  x 2 , g ( x)  6  x
b) f ( x)  x , g ( x)  9  x 2
c) f (x) 
4 x
1
, g ( x )  (omit f  f )
2
x
x 1
(If you need more practice on this topic, do p. 67 #1-2)
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Math 31
Pre-Calculus
Rationalizing
Recall:
 Rationalizing a denominator means to write an equivalent expression without a radical in
the denominator. (We can also rationalize numerators.)
 We do this by multiplying the numerator and denominator by the _________________
_________________ e.g. x  a x  a  x 2  a
 Sometimes this allows us to simplify the rational expression.



Class Examples:
Rationalize the denominator and simplify
Rationalize the numerator and simplify
x 2  16
x 2
.
7  x2  2
.
x3
7. Rationalize the denominator and simplify:
x 5
a)
x 5
b)
8
4x
2 x  2
Math 31
Pre-Calculus
8. Rationalize the numerator and simplify:
3 x 5
x 3
a)
b)
x 2  16
x 9
(If you need more practice on this topic, do p. 4 #1-2)
Once you can do all the math in this booklet, you should be ready to start calculus!
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