Download Warm Up Introduction to Polynomials and Adding and Subtracting

Warm Up
Introduction to Polynomials and
Adding and Subtracting
Polynomials
Classifying Polynomials:
We classify polynomials based on the number of
_________.
terms
What is a monomial? A polynomial with only 1 term.
What is a binomial? A polynomial with 2 terms.
What is a trinomial? A polynomial with 3 terms.
When do we use the term polynomial?
We use the term polynomial to describe any expression
which contains some combination of variables and numbers.
What is a term?
Terms are made up of numbers, variables or
the product and/or quotient of some
combination of numbers and variables.
EXAMPLE: 2x is a term. (the product of a # and variable)
2x/3 is a term. (the quotient of a # and variable)
+
Terms are separated by ________ and ________
signs!
-
Classify each of the polynomials
1.
2.
3.
4.
3x2y + 3x + 4y
Trinomial
2x2y4z
Monomial
2xy2 + 3x2y + 9x Trinomial
5x2y + 2xy3
Binomial
Calculating the degree of a
polynomial..
If it is a single variable term, find the term
with the largest exponent.
5x3 + 9x2 + x1
Remember the
This is a 3rd degree polynomial.
understood 1!!!
If it is a multi-variable term, you add the
exponents of all of the variables.
13+9
4x2y1+ 2xy
This is a 4th degree polynomial (3+1=4). Don’t forget
the understood 1’s!!!!
Calculate the degree of the
polynomial…
1. 6x2 + 4x + 4x3 + x
3rd degree
2
4
2
4
2. 21x y + 12x y + 2xy
6th degree
2
3
3. 3x + 2y
3rd degree
Writing a polynomial in standard
form…
•
Standard form is in descending order of
the x variable.
1. 2xy3 + 3x2y + x3
X3 + 3x2y + 2xy3
2. 3xy2 + 3x3 + 2x2y
3
2
2
3x + 2x y + 3xy
Adding and Subtracting
Polynomials…
is nothing more than combining like terms.
Remember you can do this vertically or
horizontally.
1. (2x2 - 3x + 3) + (4x2 + 5x - 9) 6x2 + 2x -6
2. (3x2 - 9x - 5) - (-2x2 - 4x + 5) 2
5x - 5x - 10
5(x2 - x - 2)
Ex 1 (x2 + 3x + 4) + (-2x2 + 10x - 5)
Combine LIKE terms
x2 + 3x + 4
-2x2 + 10x - 5
-1x2 + 13x - 1
Final Answer
Ex 2. (4b3 - 2b) + (b3 + 6b2 + 3b - 7)
4b3 + 0b2 - 2b + 0
b3 + 6b2 + 3b - 7
5b3 + 6b2 + 1b - 7
Final Answer
Ex 3. (12y2 - 8y + 4) - (9y2 + 5y + 1)
Don’t forget to Distribute the -1!!
(12y2 - 8y + 4) - 1(9y2 + 5y + 1)
(12y2 - 8y + 4) - 9y2 - 5y - 1
12y2 - 8y + 4
- 9y2 - 5y - 1
3y2 - 13y + 3
Final Answer
Ex 4. (3a3 + 10a - 15) - (-a3 + 2a2 + 6a - 9)
(3a3 + 10a - 15) - 1(-a3 + 2a2 + 6a - 9)
3a3 + 10a - 15 + a3 - 2a2 - 6a + 9
3a3 + 0a2 + 10a - 15
+a3 - 2a2 - 6a + 9
4a3 - 2a2 + 4a - 6
Final Answer
POLYNOMIAL MULTIPLICATION
(2x - 3)(x + 5)
3 methods
a. Distribute (2x - 3)(x + 5)
*4 multiplications
2x2 + 10x - 3x - 15 (then combine like
terms)
2x2 + 10x - 3x - 15
2x2 + 7x - 15
Final Answer
b. 3rd grade style
2x - 3
x+5
+10x - 15
2x2 - 3x + 0
2x2 + 7x - 15
Final
Answer!
c. Box Method
2x
x 2x2
+ 5 +10x
-3
-3x
-15
2x2 + 7x - 15
Final Answer
Ex 5. (4x + 7)(3x + 7)
a.) 12x2 + 28x + 21x + 49
12x2 + 49x + 49
b.)
4x + 7
3x + 7
+28x + 49
12x2 + 21x + 0
12x2 + 49x + 49
c.)
4x + 7
3x 12x2 +21x
+ 7 +28x + 49
Ex 6. (x - 1)(x2 - 4x + 6)
a. x3 - 4x2 + 6x - 1x2 + 4x - 6
x3 - 5x2 + 10x - 6
b.
x2 - 4x + 6
x -1
-1x2 + 4x - 6
1x3 -4x2 + 6x + 0
x3 - 5x2 + 10x - 6
c.
x2 - 4x + 6
x x3 -4x2 +6x
-1 -1x2 +4x - 6
x3 - 5x2 + 10x - 6
Ex 7. (2z2 + 3z - 4)(4z + 5)
a. 8z3 + 10z2 + 12z2 + 15z - 16z - 20
8z3 + 22z2 - 1z - 20
2z2 + 3z - 4
4z + 5
+10z2 + 15z - 20
8z3 +12z2 - 16z + 0
8z3 + 22z2 - 1z - 20
b.
c. 2z2 + 3z - 4
4z 8z3 +12z2 -16z
+5 10z2 +15z - 20
Ex 8. (2a + 5)(2a - 5)
a. 4a2 - 10a + 10a - 25
4a2 - 25
b.
2a + 5
2a - 5
-10a - 25
4a2 +10a + 0
4a2 - 2 5
c.
2a +5
2a 4a2 +10a
-5 -10a -25
4a2 - 2 5
Ex 9. (3m + 4n)2
a. (3m + 4n)(3m + 4n)
9m2 + 12mn + 12mn + 16n2
9m2 + 24mn + 16n2
b.
3m + 4n
c.
3m
+4n
3m + 4n
3m 9m2 +1 2mn
+12mn + 16n2
+4n +12mn +16n2
9m2 + 12mn + 0
9m2 + 24mn + 16n2
9m2 + 24mn + 16n2