What do these prefixes mean?
Can you give a word that
starts with them?
MONO
BI
TRI
POLY
Term
A number, a variable, or
the product/quotient of
numbers/variables.
Examples?
Degree of a Term
The exponent of the
variable.
We will find them only
for one-variable terms.
Term
Degree of Term
3
0
4x
1
-5x2
2
18x5
5
Polynomial
A term or the
sum/difference of terms
which contain only 1
variable. The variable
cannot be in the
denominator of a term.
Degree of a
Polynomial
The degree of the term
with the highest degree.
Polynomial
Degree of
Polynomial
6x2 - 3x + 2
2
15 - 4x + 5x4
4
x + 10 + x
1
x3 + x2 + x + 1
3
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 - 4x + 5x4
5x4 - 4x + 15
x + 10 + x
2x + 10
1 + x2 + x + x 3
x3 + x2 + x + 1
Naming Polynomials
Polynomials are named
or classified by their
degree and the
number of terms they
have.
Polynomial
Degree
Degree Name
7
0
constant
5x + 2
1
linear
4x2 + 3x - 4
2
quadratic
6x3 - 18
3
cubic
For degrees higher than 3 say:
“nth degree”
x5 + 3x2 + 3
x8 + 4
“5th degree”
“8th degree”
Polynomial
# Terms
# Terms Name
7
1
monomial
5x + 2
2
binomial
4x2 + 3x - 4
3
trinomial
6x3 - 18
2
binomial
For more than 3 terms say:
“a polynomial with n terms” or “an n-term polynomial”
11x8 + x5 + x4 - 3x3 + 5x2 - 3
“a polynomial with 6 terms” – or – “a 6-term polynomial”
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
9.1 Adding and Subtracting Polynomials
Adding Polynomials
in horizontal form:
(x2 + 2x + 5) + (3x2 + x + 12)
= 4x2+ 3x + 17
in vertical form:
x2 + 2x + 5
+ 3x2 + x + 12
= 4x2+ 3x + 17
9.1 Adding and Subtracting Polynomials
Simplify: The classify the polynomial by number of
terms and degree
x2
+x+1
+ 2x2 + 3x + 2
3x2 +
4x + 3
(4b2 + 2b + 1)
+ (7b2 + b – 3)
11b2 + 3b – 2
3x2 – 4x + 8
+ 2x2 – 7x – 5
5x2 – 11x + 3
a2 + 8a – 5
+ 3a2 + 2a – 7
4a2 + 10a – 12
9.1 Adding and Subtracting Polynomials
Simplify: The classify the polynomial by number of
terms and degree
7d2 + 7d
+ 2d2 + 3d
z2 + 5z + 4
+ 2z2 - 5
9d2 + 10d
3z2 + 5z – 1
(2x2 – 3x + 5)
+ (4x2 + 7x – 2)
6x2 + 4x + 3
(a2 + 6a – 4)
+ (8a2 – 8a)
9a2 – 2a – 4
9.1 Adding and Subtracting Polynomials
Simplify: The classify the polynomial by number of
terms and degree
3d2 + 5d – 1
+ (–4d2 – 5d + 2)
+
7p2 + 5
(–5p2 – 2p + 3)
–d2+ 1
2p2 – 2p + 8
(3x2 + 5x)
(x3 + x2 + 7)
+ (4 – 6x – 2x2)
x2 – x + 4
+ (2x2 + 3x – 8)
x3 + 3x2 + 3x – 1
9.1 Adding and Subtracting Polynomials
Simplify:
2x3
x2
+
–4
+ 3x2 – 9x + 7
2x3 + 4x2 – 9x + 3
5y2 – 3y + 8
+ 4y3 – 9
4y3 + 5y2 – 3y – 1
4p2 + 5p
+ (-2p + p + 7)
4p2 + 4p + 7
(8cd – 3d + 4c)
+ (-6 + 2cd – 4d)
4c – 7d + 10cd – 6
9.1 Adding and Subtracting Polynomials
Simplify:
(12y3 + y2 – 8y + 3) + (6y3 – 13y + 5)
= 18y3 + y2 – 21y + 8
(7y3 + 2y2 – 5y + 9) + (y3 – y2 + y – 6)
= 8y3 + y2 – 4y + 3
(6x5 + 3x3 – 7x – 8) + (4x4 – 2x2 + 9)
=
6x5+ 4x4 + 3x3 – 2x2 – 7x + 1
Subtracting Polynomials
ubtraction is adding the opposite)
(5x2 + 10x + 2) – (x2 – 3x + 12)
(5x2 + 10x + 2) + (–x2) + (3x) + (–12)
= 4x2+ 13x – 10
5x2 + 10x + 2
–(x2 – 3x + 12)
= 5x2 + 10x + 2
= –x2 + 3x – 12
4x2 + 13x – 10
9.1 Adding and Subtracting Polynomials
Simplify:
(3x2 – 2x + 8) – (x2 – 4)
3x2 – 2x + 8 + (–x2) + 4
= 2x2 – 2x + 12
(10z2 + 6z + 5) – (z2 – 8z + 7)
10z2 + 6z + 5 + (–z2) + 8z + (–7)
= 9z2 + 14z – 2
9.1 Adding and Subtracting Polynomials
Simplify:
7a2
– 2a
– (5a2 + 3a)
2a2 – 5a
4x2
+ 3x + 2
– (2x2 – 3x – 7)
2x2 + 6x + 9
–
3x2 – 7x + 5
(x2 + 4x + 7)
2x2 – 11x – 2
7x2 – x + 3
– (3x2 – x – 7)
4x2 + 10
9.1 Adding and Subtracting Polynomials
Simplify:
3x2
– 2x + 10
– (2x2 + 4x – 6)
x2 – 6x + 16
3x2
– 5x + 3
– (2x2 – x – 4)
x2 – 4x + 7
2x2 + 5x
– (x2 – 3)
x2 + 5x + 3
4x2 – x + 6
– (3x2 – 4)
x2 – x + 10
9.1 Adding and Subtracting Polynomials
Simplify:
(4x5 + 3x3 – 3x – 5) – (– 2x3 + 3x2 + x + 5)
= 4x5 + 5x3 – 3x2 – 4x – 10
(4d4 – 2d2 + 2d + 8) – (5d3 + 7d2 – 3d – 9)
= 4d4 – 5d3 – 9d2 + 5d + 17
(a2 + ab – 3b2) – (b2 + 4a2 – ab)
= –3a2 + 2ab – 4b2
9.1 Adding and Subtracting Polynomials
Simplify by finding the perimeter:
6c + 3
A
c2 + 1
4c2 + 2c + 5
A=
5c2
x2 + x
+ 8c + 9
2x2
B
x2 + x
B = 6x2 +
2x
2x2
9.1 Adding and Subtracting Polynomials
Simplify by finding the perimeter:
3x2 – 5 x
x2
+7
C
2d2 + d – 4
3d2 – 5d
D
3d2 – 5d
d2 + 7
C = 8x2 – 10x +
14
D = 9d2 – 9d +
3
9.1 Adding and Subtracting Polynomials
Simplify by finding the perimeter:
3x2 – 5x
a+1
E
a3 + 2a
x2 + 3
F
2a3 + a + 3
E = 3a3 + 4a + 4
F = 8x2 – 10x +
6
HW: Section 9.1
Page 459 (2-26 even)