Download Polynomials

Polynomials
Sec 9.1.1
Learning Targets
 Vocabulary
 Operations between polynomials
 Introduction to graphs of polynomials
Definitions
 Polynomial comes from poly- (meaning "many") and
-nomial (in this case meaning "term") ... so it means
“many terms”
 Term: A number, a variable, or the product/quotient
of numbers/variables.
Polynomial
 Example of Polynomial
 𝑦 = 5𝑥 3 + 0.5𝑦 2 − 19𝑥𝑦 3 − 7
A Term has 3 Components:
5𝑥
3
Exponent:
Can only be positive
integers: 0,1,2, 3,
Coefficient: can be
any real number…
including zero.
Variable
These components
are very
important!!!
NOT ALLOWED
 Negative exponents:
𝑥 −2
 Variables in the denominator:
4
3−𝑥
Check In
 Which of the following is a polynomial:
−6𝑦2
−
2) 4𝑥 −
1
3
1)
3) 3𝑥𝑦𝑧 +
4) 5
7
9
𝑥
𝑦 + 𝑥𝑦 −3 − 41𝑥𝑦
3𝑥
𝑦2𝑧
− 0.1𝑥𝑧 − 200𝑦 + 0.5
Naming a Polynomial
 We can classify a polynomial based on how many
terms it has:
Polynomial
# Terms
# Terms Name
7
1
monomial
5x + 2
2
binomial
4x2 + 3x - 4
3
trinomial
6x3 - 18
2
binomial
Naming Cont.

Quadrinomial (4 term) and quintinomial (5 term) also exist,
but those names are not often used.

Polynomials Can Have Lots and Lots of Terms

Polynomials can have as many terms as needed, but not an
infinite number of terms.
For more than 3 terms say:
“a polynomial with n terms” or “an nterm polynomial”
11x8 + x5 + x4 - 3x3 + 5x2 - 3
“a polynomial with 6 terms” – or – “a 6term polynomial”
Degree of a Term
The degree of a term is determined by the
exponent of the variable.
Term
Degree of Term
3
0
4x
1
-5x2
2
18x5
5
Naming a Polynomial
 We can also classify a polynomial based on its highest
degree:
Polynomial
Degree
# Degree Name
7
0
Constant
5x + 2
1
Linear
4x2 + 3x - 4
2
Quadratic
6x3 - 18
3
Cubic
Putting it All Together
Polynomial
Name
-14x3
cubic monomial
-1.2x2
quadratic monomial
-1
constant monomial
7x - 2
linear binomial
3x3+ 2x - 8
cubic trinomial
2x2 - 4x + 8
quadratic trinomial
x4 + 3
4th degree binomial
Standard Form of a Polynomial
A polynomial written so that the degree of
the terms decreases from left to right and no
terms have the same degree.
Not Standard
Standard
6x + 3x2 - 2
3x2 + 6x - 2
15 - 3x - x+ 5x4
5x4 - 4x + 15
x + 10 + x
2x + 10
1 + x2 + x + x 3
x3 + x2 + x + 1
Operations
 Polynomials can be added, subtracted, multiplied
and/or divided
 The following slides will cover addition, subtraction
and multiplication
 We will learn about division later on in the unit
Adding and Subtracting Polynomials
To add or subtract polynomials, simply
combine like terms.
(5x2 - 3x + 7) + (2x2 + 5x - 7)
= 7x2 + 2x
(3x3 + 6x - 8) + (4x2 + 2x - 5)
= 3x3 + 4x2 + 8x - 13
(2x3 + 4x2 - 6) – (3x3 + 2x - 2)
(2x3 + 4x2 - 6) + (-3x3 + -2x - -2)
= -x3 + 4x2 - 2x - 4
Polynomial Multiplication
 To multiply polynomials we must distribute all of the
terms
 Ex: 𝑥 3 + 4𝑥 2 + 1 × −3𝑥 2 − 2𝑥
𝑥 3 + 4𝑥 2 + 1
−3𝑥 2
−3𝑥 5 − 12𝑥 4 − 3𝑥 2
−2𝑥
−2𝑥 4 − 8𝑥 3 − 2𝑥
−3𝑥 5 − 14𝑥 4 − 8𝑥 3 − 3𝑥 2 − 2𝑥
Polynomial Multiplication
Multiply the following polynomials:
1) x  52x  1
2) 3w  22w  5
3) 2a 2  a  12a 2  1
Polynomial Multiplication
1) x  52x  1
(x + 5)
x (2x + -1)
-x + -5
+
2x2 + 10x
2x2 + 9x + -5
(3w + -2)
2) 3w  22w  5
x (2w + -5)
-15w + 10
+
6w2 + -4w
6w2 + -19w + 10
Polynomial Multiplication
3) 2a  a  12a  1
2
2
(2a2 + a + -1)
x (2a2 + 1)
2a2 + a + -1
+
4a4 + 2a3 + -2a2
4a4 + 2a3 + a + -1
Investigating Graphs of
Polynomials
 Pg. 437
 In your notes go through problem 9-1 silently…
 Write down any conjectures, similarities or patterns you
see
 After 5 minutes we will discuss in our teams
For Tonight
 Homework:
 Pg. 440: 9-8 9-11, 9-13, 9-14 and 9-18
 Answers to these questions will be posted online
tonight