Download Notes for Chapter 5

Notes for Chapter 5
Factoring
GCF (Greatest Common Factor)
For as set of monomials find:
1. the common coefficients
2. the common variables
3. place these common parts outside parenthesis
4. divide each monomial by the common parts, placing the results inside the
parenthesis
Example:
48 x 6 y 2  40 x 4 y 3 becomes 8 x 4 y 2 (6 x 2  5 y)
Trinomial factoring (leading coefficient of 1, x 2  bx  c )
Find 2 factors of ‘c’ whose sum is ‘b’
Calculator help
c
y1 
In the tables, find ‘b’ in y 2 . Read the factors in the remaining columns.
x
y 2  x  y1
Example:
x 2  7 x  12 in factored form becomes (x+3)(x+4).
Trinomial factoring (leading coefficient ‘a’ not 1, ax 2  bx  c )
Find 2 factors of ‘a’ times ‘c’ whose sum is ‘b’
Calculator help
ac
y1 
In the tables, find ‘b’ in y 2 . Read the factors in the remaining columns.
x
y 2  x  y1
GCF
GCF
GCF
ax
fx
2
GCF
fx
c
‘f’ is for either factor of ‘ac’. Find the horizontal and vertical GCF, to find the
factors. Make the GCF negative, if the box next to it is negative.
Example:
12 x 2  7 x  10
12*(-10) = -120
Factors of 120 that add to -7 are 8 and -15.
3x
4x
-5
2
2
12x
-15x
8x
-10
Result: (4x-5)(3x+2)
Special case factoring – Difference of Squares
a2  b2
becomes
(a  b)( a  b)
Example:
x 2  64
becomes
Note that 8*8=64 (a perfect square)
( x  8)( x  8)
Special case factoring – Sum of Cubes
a3  b3
becomes
(a  b)( a 2  ab  b 2 )
Example:
x 3  64
becomes
( x  4)( x 2  4 x  16)
Note that 4*4*4-64 (a perfect cube)
Special case factoring – Difference of Cubes
a3  b3
becomes
(a  b)( a 2  ab  b 2 )
Example:
x 3  64
becomes
Note that 4*4*4-64 (a perfect cube)
( x  4)( x 2  4 x  16)
Multiplying binomials
FOIL method (First, Outside, Inside Last)
Synthetic Division – divisor coefficient of 1
Example:
x 3  2 x 2  5x  6
x3
-3] 1 2 -5 -6
____ -3_ 3 6
1 -1 -2
Result: x 2  x  2
2nd Example (missing elements of the polynomial)
x 3  125
x5
-5] 1 0 0 125
____ -5_ 25 -125
1 -5 25
Result: x 2  5 x  25
Synthetic Division – divisor coefficient not 1
Example:
2 x 3  x 2  5x  2
2x  1
Divide everything by the coefficient of the divisor - (2 in this example)
x3 

1
|
2
1 2 5
x  x 1
2
2
1
x
2
1
5
-1
2
2
1
1

1
2
2
________________
1 -1
-2
1 
Result: x 2  x  2
Complex numbers
i  1
i 2  1
i 3  i
Know the powers of i.
i4  1
Complex number convention; i is not in the denominator.
Multiply by the conjugate to remove the imaginary number from the denominator.
3i
multiply by the conjugate to simplify this fraction.
1  2i
3  i 1  2i 3  6i  i  2 5  5i
*


 1 i
1  2i 1  2i 1  2i  2i  4
5
Using the calculator, set mode to a+bi . Also, use FRAC to decipher results.
Solving equations in one variable
Use inverse operations appropriately:
Operation
Addition
Subtraction
Multiplication
Division
Squaring
Cubing
Inverse
Operation
Subtraction
Addition
Division
Multiplication
Square Root
Cube Root
Example:
5  2 x  3  12
2 x3  7
7
2
49
x3
4
49
49 12
x
3

4
4
4
61
x
 15.25
4
x3 
Be prepared to use multiple skills to solve any individual problem presented.