Download Polynomials and Factoring Review Notes

Here's the deal. The exponent is the number
of decimal places to move left or right to
remove the power of 10. How's that? Ok,
let's consider a few examples.
CHAPTER 5 REVIEW NOTES Examples :
Properties of Exponents
5 x 2 
If m and n are integers, then
5
x2
Example:
3
 3x 2
2
x
xm xn  xm n
Take the number 132.45, we want to write
this number in scientific notation.
Example: (x2)(x4) = x6
If n is an integer and b  0 , then
If m and n are integers, then
an
 a
   n
 b
b
(x )  x
m n
n
mn
5 2
10
x0  1
If m, n, and p are integers, then
If n is a positive integer (-n is negative)
, and b  0 , then
( xy )n  x n y n ,
 a
 
 b
Example: (xy)4 = x4y4
(x y )  x
n p
m p
n
 b
 
 a
n
Example :
and
y
n p
Example: (x3y2)4 = x12y8
If m and n are integers and x  0 , then
m
x
 x m n
n
x
5
x
 x5  2  x3
2
x
If n is a positive integer (-n is negative)
, and x  0 , then
 x
 
3
2

32
9

x2 x2
x
1
 n
x
and
1
 xn
n
x
consider 0.000716
The mantissa must be greater than or
equal to 1 and less than 10, and this
number is much to small. If we move the
decimal right by 1 we get 0.00716 but we
need to multiply by 10-1 to keep the same
number. So if we move the decimal right by
4 we get 7.16 which is in range for the
mantissa but we need to multiply by 10-4 to
keep the same number. So what we have is
0.000716 = 7.16 x 10-4
Multiplication/Division
This is easier, so we'll do this first.
Scientific Notation (exponential notation)
refers to writing decimal numbers as a
number multiplied by a power of 10.
Formally now, an exponential number has
two parts, a mantissa and an exponent. The
mantissa is a number whose magnitude is
greater than or equal to 1 but less than 10.
The exponent is a power of 10.
Here's an example: 2.43 x 103
the mantissa is 2.43 the exponent is 3
Here's another example: -5.104 x 10
n
(notice, we moved the decimal left twice and
increased the exponent of ten by 2)
Example:
Example: 50 = 1
Example :
So we have 132.45 = 1.3245 x 102
2
x2 x2
 x
   2 
3
9
3
If x is a real number and x  0 , then
m
We get: 1.3245 for the mantissa and this
number is in the correct range but we now
need to multiply by 100 = 102 to keep the
same number.
Example :
Example: (x ) = x
In order for the mantissa to be between 1
and 10 we will have to move the decimal
point to the left by 2 places.
-2
the mantissa is -5.104 and the exponent is -2
Ok, so what are these numbers?
2.43 x 103 = 2,430 and -5.104 x 10-2 = 0.05104
Procedure:
Multiply/divide the mantissas, then the
powers of 10, then write the result in proper
form.
Examples:
1) 8.3x104 X 2.0x105 = (8.3 x 2.0) X ( 104
x 105 ) = 16.6 x 109
now, rewriting, we get 1.66x1010
2)
1.210    1.2    10
4.0 10   4.0   10
1
1
5
5
 0.3 10 15  0.3 10 6



Rules for Variable Expressions:
Only like terms can be added, and when adding like terms, do not change the exponent of the variable.
5x2 + 3x2 = 8x2
When multiplying variable expressions, add exponents of like variables
(5xy3)(2y2)=10xy3+2 = 10xy5
When taking powers of variable expression that is a monomial (one term only, that is - there is no addition or
subtraction inside the parentheses), multiply exponents of EVERY term inside the parentheses.
(2x3y4)3 = 23x3*3y4*3 = 8x9y12
When taking powers of a variable expression that is a binomial, trinomial or some other polynomial, use the rules
of polynomial multiplication.
For example: (x+2)2 DOES NOT EQUAL x2 + 22
(x+2)2 = (x+2)(x+2) = x2+4x + 4 (FOIL METHOD)
Example 2:
(x2+3x+5)2 = (x2+3x+5)(x2+3x+5)
=(x2+3x+5)x2 + (x2+3x+5)3x + (x2+3x+5)5 (DISTRIBUTIVE PROPERTY)
POLYNOMIALS VOCABULARY
polynomial – is a term or sum of terms in which all variables have whole number exponents.
Example: 3x, or x2 + 1, or -3x2 + 3x + 1
monomial – a number, a variable, or a product of numbers and variables.
Example: 3, 2x, -4x2 are all monomials.
binomial – the sum of two monomials that are unlike terms.
trinomial – the sum of three monomials that are unlike terms.
like terms – terms of a variable expression that have the same variable and the same exponent.
Example: 3x and 3x2 are unlike terms, but 3x and 2x are like terms.
factor – (in multiplication) a number being multiplied.
Example: What are the factors of 121? 1, 11, and 121.
121 = 11 X 11, 121 = 1 X 121
to factor a polynomial – to write a polynomial as a product of other polynomials
to factor a trinomial of the form ax2 + bx + c - to express the trinomial as the product of two
binomials.
Example: x2 + 5x + 6 = (x+2)(x+3)
to factor by grouping – to group and factor terms in a polynomial in such a way that a common
binomial factor is found.
Example: 2x(x+1) – 3(x+1) = (x + 1)(2x – 3)
factor completely – to write a polynomial as a product of factors that are nonfactorable over the
integers.
FOIL method– A method of finding the product of two binomials in which the sum of the products of
the First terms, of the Outer terms, of the Inner terms, and of the Last terms is found.
Example: (x+2)(x+3) =x2+ 3x + 2x + 2*3 = x2 + 5x + 6
common factor – a factor that is common to two or more numbers.
Example: What are the common factors of 12x and 16x2?
The factors of 12x are 1,2,3,4,6,12, and x
The factors of 16x2 are 1,2,4,8,16, x, x
The common factors of 12x and 16x2 are 1x, 2x, and 4x
The Greatest Common Factor of 12x and 16x2 is 4x. Find the GCF of the coefficients and choose the variable
term with the lowest exponent. The term with the lowest exponent between x and x2 is x.
Multiplying Polynomials:
Binomials can be multiplied using the FOIL method (see above).
Also, there is the funny old man method….
(3x + 5)(2x – 1)
= (3x)(2x) + (5)(-1) + (5)(2x) + (3x)(-1)
= 6x2 -5 + 10x -3x = 6x2 – 5 + 7x
= 6x2 + 7x - 5
The BOX METHOD can be used for multiplying polynomials with more than 2 terms.
Multiply (2x - 5)(x2 - 5x + 4)
A General Strategy for Factoring a Polynomial
1. Do all the terms in the polynomial have a common factor? If so, factor out the
Greatest Common Factor. Make sure that you don’t forget it in your final answer.
Example: 24x4 - 6x2 = 6x2(4x2 - 1). Also look to see if the other polynomial factor and be factored more.
(4x2-1)=(2x-1)(2x+1), so the final answer is
24x4 - 6x2 =6x2(2x-1)(2x+1),
2. Count the number of terms in the polynomial.
Two terms:
Is it a difference of squares? Factor by using: a2-b2 = (a+b)(a-b)
Example: 36x2 – 49 = (6x)2 – 72 = (6x-7)(6x+7)
Is it a difference of cubes? Factor by using a3 – b3 = (a - b)(a2 + ab + b2)
Is it a sum of cubes? Factor by using a2 + b3 = (a + b)(a2 - ab + b2)
If the polynomial can’t be factored, it is PRIME.
Three terms: Is it a perfect square trinomial?
If it is it would be in the form a2x2 + 2abx + b2 , which is factored as (a+b)2
or a2x2 + 2abx + b2 which is factored as (a-b)2
Example: 4x2 + 12x + 9 = (2x)2 + 2(2)(3)x + 32 = (2x + 3)2
Is it of the form x2 + bx + c?
6
Can you use substitution to make it in this form?
3
Example: x – 5x + 6. Since x6 = (x3)2, we can use y = x3
New form: y2 -5y + 6
Factor by finding two numbers that multiply to c and add to b.
Example: y2 -5y + 6 = (y - 3)(y - 2) because -3*-2 = 6 and -3 + -2 = -5
Substitute y = x3 back into your expression: (x3 - 3)(x3 - 2)
Can’t find the numbers? Maybe the polynomial is PRIME.
Is it of the form ax2 + bx + c?
Try factoring by the Grouping Method (or ac Method) or Snowflake Method.
Example: 2x2 + 13x + 15
(the a*c method means multiply 2*15 which is 30.
Find factors of 30 that add up to the middle term’s coefficient, which is 13. 3*10=30 and
3+10 = 13. Split the middle term into two parts:
2x2 + 10x + 3x + 15 and then factor by grouping.
2x(x+5)+3(x+5) = (2x+3)(x+5)
Those methods don’t work? Maybe the polynomial is PRIME.
Four terms: Try Factoring by Grouping. Group the 1st two terms and the last two terms. Factor out the
Greatest Common Factor from each grouping. Then factor out the common binomial term.
3. Always factor completely. Double check that each of your factors can not be factored more.
Example: (x + 2)(x2 – 1) can be factored further since (x2 – 1) is the difference of perfect squares:
(x + 2)(x2 – 1) = (x + 2)(x+1)(x-1)
4. Check your work by multiplying the factors together. Does it result in the original polynomial?