Transcript Polynomials: Terms & Factoring

Polynomials
Terms and Factoring
Algebra I H.S.
Created by:
Buddy L. Anderson
Vocabulary
 Monomial:
A number, a variable or the product of a
number and one or more variables
 Polynomial: A monomial or a sum of monomials.
 Binomial: A polynomial with exactly two terms.
 Trinomial: A polynomial with exactly three terms.
 Coefficient: A numerical factor in a term of an
algebraic expression.
 Degree of a monomial: The sum of the exponents of
all of the variables in the monomial.
Vocabulary
 Degree
of a polynomial in one variable: The largest
exponent of that variable.
 Standard form: When the terms of a polynomial are
arranged from the largest exponent to the smallest
exponent in decreasing order.
Degree of a Monomial
 What
 The
is the degree of the monomial?
4 2
5x b
degree of a monomial is the sum of the
exponents of the variables in the monomial.
 The exponents of each variable are 4 and 2. 4+2=6.
 Therefore the degree is six and it can be referred to
as a sixth degree monomial.
Polynomial
A
polynomial is a monomial or the sum of monomials
 Each monomial in a polynomial is a term of the
polynomial.
The
number factor of a term is called the
coefficient.
The coefficient of the first term in a polynomial is
the lead coefficient
 A polynomial with two terms is called a binomial.
 A polynomial with three terms is called a trinomial.
Degree of a Polynomial in One
Variable
 The
degree of a polynomial in one variable is the
largest exponent of that variable.
5x  2 x  14
2
 The
degree of this polynomial is 2, since the highest
exponent of the variable x is 2.
Standard Form of a Polynomial
 To
rewrite a polynomial in standard form, rearrange
the terms of the polynomial starting with the largest
degree term and ending with the lowest degree term.
 The leading coefficient, the coefficient of the first
term in a polynomial written in standard form, should
be positive.
Put in Standard Form
7  3x  2 x
3
2
 3x  2x  7
3
2
13x  2x  7
3
2
3x 3  2 x 2  7

Factoring Polynomials
By
Grouping
Difference of Squares
Perfect Square Trinomials
X-Box Method
By Grouping
When
polynomials contain four terms, it
is sometimes easier to group like terms in
order to factor.
Your goal is to create a common factor.
You can also move terms around in the
polynomial to create a common factor.
By Grouping
FACTOR: 3xy - 21y + 5x – 35
Factor the first two terms:
3xy - 21y = 3y (x – 7)
Factor the last two terms:
+ 5x - 35 = 5 (x – 7)
The terms in the parentheses are the same so
it’s the common factor
Now you have a common factor
(x - 7) (3y + 5)
By Grouping
FACTOR: 15x – 3xy + 4y –20
Factor the first two terms:
15x – 3xy = 3x (5 – y)
Factor the last two terms:
+ 4y –20 = 4 (y – 5)
The terms in the parentheses are opposites so
change the sign on the 4
- 4 (-y + 5) or – 4 (5 - y)
Now you have a common factor (5 – y) (3x – 4)
Difference of Squares
When
factoring using a difference of squares,
look for the following three things:
only 2 terms
minus sign between them
both terms must be perfect squares
If all 3 of the above are true, write two
( ), one with a + sign and one with a – sign :
( + ) ( - ).
Try These
1.
2.
3.
4.
5.
6.
a2 – 16
x2 – 25
4y2 – 16
9y2 – 25
3r2 – 81
2a2 + 16
Perfect Square Trinomials
When
factoring using perfect square
trinomials, look for the following three things:
3 terms
last term must be positive
first and last terms must be perfect squares
If all three of the above are true, write one (
)2 using the sign of the middle term.
Try These
1.
2.
3.
4.
5.
6.
a2 – 8a + 16
x2 + 10x + 25
4y2 + 16y + 16
9y2 + 30y + 25
3r2 – 18r + 27
2a2 + 8a - 8
X-Box Method
No,
we are not going to feed
polynomials into a game system that
will factor them.
We will go over the following
example.
X-Box Method
(3)(-10)=
-30
2
-15
x
-5
3x
3x2
-15x
+2
2x
-10
-13
3x2 -13x -10 = (x-5)(3x+2)
X-Box Method
The
color codes in the equation show
where the numbers go in the diamond
and box.
The -15 and 2 came from the fact
that you needed 2 numbers that
multiplied to get -30 and added to get
-13.
X-Box Method
The
outside of the box are the GCF
of what they are above or beside.
These give you you r 2 factors.