9.0 Classifying Polynomials

Objective: SWBAT describe and write polynomials in standard form. Concept:

Unit 9 – Polynomials and Factoring

MONOMIALA number, a variable, or the product/quotient of numbers and variables4 is a monomial3x is a monomial14x6 is a monomial

Degree of a Monomial

The exponent of the variable.

Monomial

3

4x

-5x2

18x5

Degree of Monomial

0

1

2

5

Degree of a Monomialwith more than one variable…

The sum of the exponents of all of its

variables.7xyz = degree 3

PolynomialA monomial or the sum/difference of

monomials 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;

what is the highest exponent??

Polynomial

6x2 - 3x + 2

15 - 4x + 5x4

x + 10 + x

x3 + x2 + x + 1

Degree ofPolynomial

2

4

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

6x + 3x2 - 2

15 - 4x + 5x4

x + 10 + x

1 + x2 + x + x3

Standard

3x2 + 6x - 2

5x4 - 4x + 15

2x + 10

x3 + x2 + x + 1

Naming PolynomialsPolynomials are named

or classified by their degree and the

number of terms they have.

Polynomial Degree Name using Degree

0 Constant

1 Linear

2 Quadratic

3 Cubic

4 Fourth term

> 4 Nth term

Polynomial

75x + 2

4x2 + 3x - 46x3 - 18

Degree

0123

Degree Name

constantlinear

quadraticcubic

For degrees higher than 3 say: “nth degree”

x5 + 3x2 + 3 “5th degree”x8 + 4 “8th degree”

Polynomial # of Terms Name using # of Terms

1 Monomial

2 Binomial

3 Trinomial

> 3 Nth term polynomial

Polynomial

75x + 2

4x2 + 3x - 46x3 - 18

# Terms

1232

# Terms Name

monomialbinomialtrinomialbinomial

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-14x3

-1.2x2

-17x - 2

3x3+ 2x - 82x2 - 4x + 8

x4 + 3

Namecubic monomial

quadratic monomialconstant monomial

linear binomialcubic trinomial

quadratic trinomial4th degree binomial

9.1 Adding and Subtracting Polynomials

Objective: SWBAT add and subtract complex polynomials.

Concept: Unit 9 – Polynomials and Factoring

Adding Polynomials

= 4x2+ 3x + 17

9.1 Adding and Subtracting Polynomials

(x2 + 2x + 5) + (3x2 + x + 12)in horizontal form:

x2+ 2x + 53x2 + x + 12+

= 4x2+ 3x + 17

in vertical form:

Simplify: Then 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

9.1 Adding and Subtracting Polynomials

3x2 – 4x + 8+ 2x2 – 7x – 5

5x2 – 11x + 3

a2 + 8a – 5 + 3a2 + 2a – 7

4a2 + 10a – 12

Simplify: Then classify the polynomial by number of terms and degree

7d2 + 7d+ 2d2 + 3d

9d2+ 10d

(2x2 – 3x + 5)+ (4x2 + 7x – 2)

6x2+ 4x + 3

9.1 Adding and Subtracting Polynomials

z2 + 5z + 4+ 2z2 - 5

3z2 + 5z – 1

(a2 + 6a – 4) + (8a2 – 8a)

9a2 – 2a – 4

Simplify: Then classify the polynomial by number of terms and degree

3d2 + 5d – 1 + (–4d2 – 5d + 2)

–d2+ 1

(3x2 + 5x)+ (4 – 6x – 2x2)

x2 – x + 4

9.1 Adding and Subtracting Polynomials

7p2 + 5+ (–5p2 – 2p + 3)

2p2 – 2p + 8

(x3 + x2 + 7) + (2x2 + 3x – 8) x3 + 3x2 + 3x – 1

Simplify: 2x3 + x2 – 4

+ 3x2 – 9x + 7 2x3 + 4x2 – 9x + 3

5y2 – 3y + 8 + 4y3 – 9 4y3 + 5y2 – 3y – 1

9.1 Adding and Subtracting Polynomials

4p2 + 5p+ (-2p + p + 7)

4p2 + 4p + 7

(8cd – 3d + 4c) + (-6 + 2cd – 4d)

4c – 7d + 10cd – 6

Simplify: (12y3 + y2 – 8y + 3) + (6y3 – 13y + 5)

(7y3 + 2y2 – 5y + 9) + (y3 – y2 + y – 6)

(6x5 + 3x3 – 7x – 8) + (4x4 – 2x2 + 9)

= 18y3 + y2 – 21y + 8

9.1 Adding and Subtracting Polynomials

= 8y3 + y2 – 4y + 3

= 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)

5x2 + 10x + 2

4x2+ 13x – 10

–(x2 – 3x + 12)

= 4x2+ 13x – 10

= 5x2 + 10x + 2 = –x2 + 3x – 12

Simplify:9.1 Adding and Subtracting Polynomials

= 2x2 – 2x + 12

(3x2 – 2x + 8) – (x2 – 4) 3x2 – 2x + 8 + (–x2) + 4

(10z2 + 6z + 5) – (z2 – 8z + 7)10z2 + 6z + 5 + (–z2) + 8z + (–7)

= 9z2 + 14z – 2

Simplify:7a2 – 2a

2a2 – 5a – (5a2 + 3a)

9.1 Adding and Subtracting Polynomials

4x2 + 3x + 2

2x2 + 6x + 9 – (2x2 – 3x – 7)

3x2 – 7x + 5

2x2 – 11x – 2 – (x2 + 4x + 7)

7x2 – x + 3

4x2 + 10 – (3x2 – x – 7)

Simplify:3x2 – 2x + 10

x2 – 6x + 16 – (2x2 + 4x – 6)

9.1 Adding and Subtracting Polynomials

3x2 – 5x + 3

x2 – 4x + 7 – (2x2 – x – 4)

2x2 + 5x

x2 + 5x + 3 – (x2 – 3)

4x2 – x + 6

x2 – x + 10 – (3x2 – 4)

Simplify:(4x5 + 3x3 – 3x – 5) – (– 2x3 + 3x2 + x + 5)

= 4x5 + 5x3 – 3x2 – 4x – 10

9.1 Adding and Subtracting Polynomials

(4d4 – 2d2 + 2d + 8) – (5d3 + 7d2 – 3d – 9)= 4d4 – 5d3 – 9d2 + 5d + 17

(a2 + ab – 3b2) – (b2 + 4a2 – ab)= –3a2 + 2ab – 4b2

Simplify by finding the perimeter:

x2 + x

2x2 B6c + 3

4c2 + 2c + 5

c2 + 1

B = 6x2 + 2x

A = 5c2 + 8c + 9

A

9.1 Adding and Subtracting Polynomials

2x2

x2 + x

3x2 – 5 x

x2 + 7 C D

2d2 + d – 4

d2 + 7

3d2 – 5d

C = 8x2 – 10x + 14

D = 9d2 – 9d + 3

3d2 – 5d

9.1 Adding and Subtracting PolynomialsSimplify by finding the perimeter:

x2 + 3 Fa + 1

2a3 + a + 3

a3 + 2a

F = 8x2 – 10x + 6

E = 3a3 + 4a + 4

E

9.1 Adding and Subtracting Polynomials

3x2 – 5x

Simplify by finding the perimeter: