Download Polynomials and Polynomial Operations

Polynomials and Polynomials
Operations
Grade 9 Math
Bedford Junior High
Polynomials:
Monomials, Binomials, and Trinomials
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A monomial is a
constant or numerical
coefficient, a variable or
literal coefficient , or the
product of a constant
and one or more
variables.
A polynomial is the sum
of one or more
monomials.
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"... this means that a
monomial is a single
term which has NO
positive sign (+) or
negative sign (-)
between entries. It
would be things like:
13, 3x, -57, x², 4y²,
-2xy, or 520x²y²
but NOT (2x+7) or (4x²2x)."
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"A polynomial can be one monomial or a number of
monomials grouped together with positive signs and/or
negative signs.”
The expression 3x² is a monomial because it has one
term.
The expression 5x² + 2x is a binomial because it has
two terms.
The expression 6x² +7x + 8 is a trinomial because it
has three terms.
All three expressions can be called polynomials.
Operations with Polynomials
Adding
Add like terms by adding
the numerical
coefficients.
ex. 4ab + -2ab = 2ab
Subtracting
 Subtract one monomial
from another like
monomials, add the
opposite of the second
number.
ex. 4ab - (-2ab) =
4ab +(+2ab)=6ab
ex. 6x2 - x2 = 5x2
6x2 +(- x2) = 5x2
Operations with Polynomials
Multiply monomials:
Multiply the numerical
coefficients.
When variable factors are
powers with the same
base, multiply by adding
exponent
ex. 3x2(4x3) = 12x5
Because (3)(4) = 12
and ( x2)(x3)
= x (2+3) = x5
Therefore
3x2(4x3) = 12x5
Divide monomials:
Divide their numerical
coefficients.
When variable factors are
powers with the same
base, divide by
subtracting exponents.
ex. 30x21
(6x4) = 5x17
( 30 ÷ 6) = 5 and
x21 ÷ x4 = x17
Therefore, the answer is 5x17
Distributive Property
Means that what is inside the bracket is
affected by the outside term.
 For example with a sharing model:
3 ( x + 2) this means we have 3 groups
of ( x+ 2) Which would give us a total of
3x +6
v
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Distributive Property
Area Model
 Remember that Area = length X width
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3( x +2) the (l) is 3 and (w) is (x+2)
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Therefore 3( x +2) = 3x + 6
Distributive Property
Symbolically :
The “3” affects each item inside the bracket. It does not matter how
many terms are inside the bracket. Each term is affected
3 ( x+ 2)
3(x) + 3(2)
3x + 6
Another example is 7x( 2x + 4y):
(7x) (2x) + (7x) (4y)
(7)(2)(x)(x) + (7)(4)(x)(y)
14 x2 + 28xy
Factoring Polynomial Expressions
Symbolically ( Finding the GCF)
Let’s say we have the following expression
35 a3 - 28 a2 + 21a
we have to look for the greatest numerical coefficient
and literal coefficient that can come out of each part
of the expression.
(7·5·a·a·a) - (4·7·a·a) + (3·7·a)
Therefore, we find the factors of each term. “7” is
common to all three terms as well as “a”. What is left
over from the term goes inside the bracket. Our GCF
is 7a . The leftovers are ( 5a2 - 4a + 3) . So our two
factors are: ( 7a) ( 5a2 - 4a + 3)
Factoring :
We can use the distributive principle and the
area model to check our answer. Our length is
(7a) and our width is ( 5a2 - 4a + 3) .
Therefore our area is 35 a3 - 28 a2 + 21a
We are now back to where we started.
7a
5a2
35a3
- 4a
-28a2
3
+21a
Multiplying Two Binomials
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When building a rectangular
tile model, use the following
guidelines:
Begin at the top left corner
with the x2 tiles ( Region 1)
Construct a rectangle in the
bottom right corner with the
unit tiles ( Region 4)
Fill the top right and bottom
left with the x-tiles
( regions 2 and 3)
If you draw a rectangle with a
length of (x + 1) and width of
( x + 5) the area of the
rectangle is :
x2 +6x +5
Region 1
Region 2
x2
X tiles
Region 3
x tiles
Region 4
Unit tiles
Multiplying Two Binomials Using the
Distributive Principle
Using the distributive principle you can
do this symbolically as follows:
 (x+3)(x+5)
 (x)(x+5) +3(x+5)
 (x)(x) + (x)(5) + (3)(x) + (3)(5)
 x2 + 5x +3x + 15 ( Remember to combine like terms)
 x2 + 8x + 15
( final simplified version)
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Another Way to Remember the
Distributive Principle
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(x+3)(x+5)
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Multiply the FIRST terms in
each bracket
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Multiply the first term in the first
set of brackets and the last term
in the section set of brackets
 (x) (5)
(OUTSIDE)
 multiply the last term in the first
set of brackets and first term in
the second set of brackets
(INSIDE)
 multiply the last terms in each
set of brackets. ( LAST)
 An acronym is FOIL
( x) ( x) = x2
=
5x
= 3x
 (3)(x)
 ( 3) (5) = 15
 Therefore we get
 x2 + 8x + 15
Multiplying Binomials with Negatives
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If you have one side of the
rectangle positive and the
other side negative, then
using the distributive principle
(x -2) ( x +3)
X2 + 3x -2x -6
You need the pieces to
complete all regions of your
rectangle , but your final
outcome will be :
X2 + x - 6 because you use
the ZERO Principle
However, to have a complete
rectangle , you NEED the
pieces + 2x and -2x
Factoring Trinomials
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When you are factoring trinomials , you
are organizing your tiles into a rectangle .
Its side measures become your factors.
One way to help you find your side
measures is to look at your unit tiles
If we were factoring
X2 +7x +6
We need to find two numbers whose sum
is 7 and whose product is 6.
We list the pairs of factors of 6. We add
each pair of factors .
Because we know that
x times x
is
x2,
then our first part of the term in
each bracket is ( x
) (x
)
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We look for the factors that have the sum
of 7 . They are 1 and 6.
We write these numbers as the second
terms in the binomials.
X2 +7x +6 = ( x + 1 ) ( x + 6)
Factors of 6
Sum of the
Factors
1,6
-1,-6
2,3
-2,-3
7
-7
5
-5
More Factoring
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If it was X2 +x -6 then it would
have to be
( x - 2 ) ( x + 3) because -2 time 3
equals -6 and -2 plus 3 equals +1
One side of your rectangle must
be positive and the other side
must be negative.
Two positive and two negative create
Zero , so we will have to put in two
positive x’s and two negative x’s to
complete the rectangle.
Factors of - Sum of the
6
Factors
-1,6
+5
1, -6
-5
-2 , 3
+1
2,-3
-1
Dividing Trinomials
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If you are dividing a trinomial ,
then you can use the area
model to help you find your
answer.
If you have x2 + 4x + 4 and
you are dividing it by (x + 2) ,
you make one of your side
measures (x + 2).
Now you can start organizing
your tiles into a rectangle .
If one side is (x+2) then the
other side MUST be (x+2)
x2 + 4x + 4 = (x+2)(x+2)
( x + 2)
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=
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( x + 2)