9.0 Classifying Polynomials Objective: SWBAT describe and write polynomials in standard form. Concept: Unit 9 – Polynomials and Factoring

MONOMIAL A number, a variable, or the product/quotient of numbers and variables 4  is a monomial 3x  is a monomial 14x 6  is a monomial

Degree of a Monomial The exponent of the variable.

Monomial 3 4x -5x 2 18x 5 Degree of Monomial

Degree of a Monomial with more than one variable… The sum of the exponents of all of its variables. 7xyz = degree 3

Polynomial A 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 6x 2 - 3x x + 5x 4 x x x 3 + x 2 + x + 1 Degree of Polynomial

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 + 3x x + 5x 4 x x 1 + x 2 + x + x 3 Standard 3x 2 + 6x - 2 5x 4 - 4x x + 10 x 3 + x 2 + x + 1

Naming Polynomials Polynomials are named or classified by their degree and the number of terms they have.

PolynomialDegreeName using Degree 0Constant 1Linear 2Quadratic 3Cubic 4Fourth term > 4Nth term

Polynomial 7 5x + 2 4x 2 + 3x - 4 6x Degree Degree Name constant linear quadratic cubic For degrees higher than 3 say: “n th degree” x 5 + 3x “5 th degree” x “8 th degree”

Polynomial# of TermsName using # of Terms 1Monomial 2Binomial 3Trinomial > 3Nth term polynomial

Polynomial 7 5x + 2 4x 2 + 3x - 4 6x # Terms # Terms Name monomial binomial trinomial binomial For more than 3 terms say: “a polynomial with n terms” or “an n-term polynomial” 11x 8 + x 5 + x 4 - 3x 3 + 5x “a polynomial with 6 terms” – or – “a 6-term polynomial”

Polynomial -14x x 2 7x - 2 3x 3 + 2x - 8 2x 2 - 4x + 8 x Name cubic monomial quadratic monomial constant monomial linear binomial cubic trinomial quadratic trinomial 4 th degree binomial

9.1 Adding and Subtracting Polynomials Objective: SWBAT add and subtract complex polynomials. Concept: Unit 9 – Polynomials and Factoring

Adding Polynomials = 4x 2 + 3x Adding and Subtracting Polynomials (x 2 + 2x + 5) + (3x 2 + x + 12) in horizontal form: x 2 + 2x x 2 + x = 4 x 2 + 3x + 17 in vertical form:

Simplify: Then classify the polynomial by number of terms and degree x 2 + x x 2 + 3x + 2 3x 2 + 4x + 3 (4b 2 + 2b + 1) + (7b 2 + b – 3) 11b 2 + 3b – Adding and Subtracting Polynomials 3x 2 – 4x x 2 – 7x – 5 5x 2 – 11x + 3 a 2 + 8a – 5 + 3a 2 + 2a – 7 4a a – 12

Simplify: Then classify the polynomial by number of terms and degree 7d 2 + 7d + 2d 2 + 3d 9d d (2x 2 – 3x + 5) + (4x 2 + 7x – 2) 6x 2 + 4x Adding and Subtracting Polynomials z 2 + 5z z z 2 + 5z – 1 (a 2 + 6a – 4) + (8a 2 – 8a) 9a 2 – 2a – 4

Simplify: Then classify the polynomial by number of terms and degree 3d 2 + 5d – 1 + ( – 4d 2 – 5d + 2) – d (3x 2 + 5x) + (4 – 6x – 2x 2 ) x 2 – x Adding and Subtracting Polynomials 7p (–5p 2 – 2p + 3) 2p 2 – 2p + 8 (x 3 + x 2 + 7) + (2x 2 + 3x – 8) x 3 + 3x 2 + 3x – 1

Simplify: 2x 3 + x 2 – 4 + 3x 2 – 9x + 7 2x 3 + 4x 2 – 9x + 3 5y 2 – 3y y 3 – 9 4y 3 + 5y 2 – 3y – Adding and Subtracting Polynomials 4p 2 + 5p + (-2p + p + 7) 4p 2 + 4p + 7 (8cd – 3d + 4c) + (-6 + 2cd – 4d) 4c – 7d + 10cd – 6

Simplify: (12y 3 + y 2 – 8y + 3) + (6y 3 – 13y + 5) (7y 3 + 2y 2 – 5y + 9) + (y 3 – y 2 + y – 6) (6x 5 + 3x 3 – 7x – 8) + (4x 4 – 2x 2 + 9) = 18y 3 + y 2 – 21y Adding and Subtracting Polynomials = 8y 3 + y 2 – 4y + 3 = 6x 5 + 4x 4 + 3x 3 – 2x 2 – 7x + 1

Subtracting Polynomials ubtraction is adding the opposite) (5x x + 2) – (x 2 – 3x + 12) (5x x + 2) + (–x 2 ) + (3x) + (–12) 5x x + 2 4x x – 10 –(x 2 – 3x + 12) = 4x x – 10 = 5x x + 2 = –x 2 + 3x – 12

Simplify: 9.1 Adding and Subtracting Polynomials = 2x 2 – 2x + 12 (3x 2 – 2x + 8) – (x 2 – 4) 3x 2 – 2x (–x 2 ) + 4 (10z 2 + 6z + 5) – (z 2 – 8z + 7) 10z 2 + 6z (–z 2 ) + 8z + (–7) = 9z z – 2

Simplify: 7a 2 – 2a 2a 2 – 5a – (5a 2 + 3a) 9.1 Adding and Subtracting Polynomials 4x 2 + 3x + 2 2x 2 + 6x + 9 – (2x 2 – 3x – 7) 3x 2 – 7x + 5 2x 2 – 11x – 2 – (x 2 + 4x + 7) 7x 2 – x + 3 4x – (3x 2 – x – 7)

Simplify: 3x 2 – 2x + 10 x 2 – 6x + 16 – (2x 2 + 4x – 6) 9.1 Adding and Subtracting Polynomials 3x 2 – 5x + 3 x 2 – 4x + 7 – (2x 2 – x – 4) 2x 2 + 5x x 2 + 5x + 3 – (x 2 – 3) 4x 2 – x + 6 x 2 – x + 10 – (3x 2 – 4)

Simplify: (4x 5 + 3x 3 – 3x – 5) – (– 2x 3 + 3x 2 + x + 5) = 4x 5 + 5x 3 – 3x 2 – 4x – Adding and Subtracting Polynomials (4d 4 – 2d 2 + 2d + 8) – (5d 3 + 7d 2 – 3d – 9) = 4d 4 – 5d 3 – 9d 2 + 5d + 17 (a 2 + ab – 3b 2 ) – (b 2 + 4a 2 – ab) = –3a 2 + 2ab – 4b 2

Simplify by finding the perimeter: x 2 + x 2x 2 B 6c + 3 4c 2 + 2c + 5 c B = 6x 2 + 2x A = 5c 2 + 8c + 9 A 9.1 Adding and Subtracting Polynomials 2x 2 x 2 + x

3x 2 – 5 x x C D 2d 2 + d – 4 d d 2 – 5d C = 8x 2 – 10x + 14 D = 9d 2 – 9d + 3 3d 2 – 5d 9.1 Adding and Subtracting Polynomials Simplify by finding the perimeter:

x F a + 1 2a 3 + a + 3 a 3 + 2a F = 8x 2 – 10x + 6 E = 3a 3 + 4a + 4 E 9.1 Adding and Subtracting Polynomials 3x 2 – 5x Simplify by finding the perimeter: