Lesson 2.1Lesson 2.1

Adding and Subtracting Polynomials.Adding and Subtracting Polynomials.

Standard and EQStandard and EQ

Standard MM1A2cStandard MM1A2c: Add, subtract, : Add, subtract, multiply and divide polynomials.multiply and divide polynomials.

Essential Question-Essential Question- What is a What is a polynomial and how do I add and polynomial and how do I add and subtract them?subtract them?

VocabularyVocabulary

MonomialMonomial- a number, a variable, or the - a number, a variable, or the product of a number and one or more product of a number and one or more variables with whole number exponents.variables with whole number exponents.

Examples: 12, xExamples: 12, x22, 15x, 15x44, 2a, 2a33bb44

VocabularyVocabulary

Degree of a MonomialDegree of a Monomial- The sum of the - The sum of the exponents of the variables in the exponents of the variables in the monomial. (The degree of a nonzero monomial. (The degree of a nonzero constant term is 0.)constant term is 0.)

Example: xExample: x22 has a degree of has a degree of 22.. 15x15x44 has a degree ofhas a degree of 44..

2a2a33bb44 has a degree of has a degree of 77.. 13 has a degree of 13 has a degree of 00..

VocabularyVocabulary

PolynomialPolynomial- a monomial or sum of - a monomial or sum of monomials, each called a monomials, each called a termterm of a of a polynomial.polynomial.

Example: 2xExample: 2x44+3x+3x22-6 is a polynomial with 3 -6 is a polynomial with 3 terms.terms.

VocabularyVocabulary

Degree of a PolynomialDegree of a Polynomial- The greatest - The greatest degree of its terms.degree of its terms.

Example: 2xExample: 2x44+3x+3x22-6 has a degree of -6 has a degree of 44..

VocabularyVocabulary

Leading CoefficientLeading Coefficient- The coefficient of the - The coefficient of the first term of a polynomial when it is written first term of a polynomial when it is written in decreasing order from left to right.in decreasing order from left to right.

Example: Example: 22xx44+3x+3x22-6 has a leading coefficient of -6 has a leading coefficient of 22..

VocabularyVocabulary

BinomialBinomial- A polynomial with two terms.- A polynomial with two terms.

Examples: 2xExamples: 2x44+3x+3x22

3x3x22-6 -6

TrinomialTrinomial- A polynomial with three terms.- A polynomial with three terms.

Examples: 2xExamples: 2x44+3x+3x22-6-6 1010xx44+5x+5x22-13-13 2x2x44+3x+3x22-6-6

Rewriting a PolynomialRewriting a Polynomial

Write 12xWrite 12x33-15x+13x-15x+13x55 so that the so that the exponents decrease from left to right. exponents decrease from left to right. Identify the degree and the leading Identify the degree and the leading coefficient.coefficient.

Degree is 5Degree is 5

13x13x55+12x+12x33-15x -15x

Leading coefficient is 13Leading coefficient is 13

Rewriting PolynomialsRewriting Polynomials

Rewrite the following so that exponents Rewrite the following so that exponents decrease from left to right. Identify the decrease from left to right. Identify the degree and leading coefficient.degree and leading coefficient.

9-2x9-2x22

16+3y16+3y33+2y+2y 6z6z33+7z+7z22-3z-3z55

4ab+5a4ab+5a22bb22-8a-8a22bb 4y+3xy+44y+3xy+4

Adding PolynomialsAdding Polynomials

Find the sum.Find the sum. (3x(3x44-2x-2x33+5x+5x22)+(7x)+(7x22+9x+9x33-2x)-2x)

Vertical format: Align like terms vertically.Vertical format: Align like terms vertically.

3x3x44-2x-2x33+5x+5x22

+ + 9x 9x33+7x+7x22-2x-2x

3x3x44+7x+7x33+12x+12x22-2x-2x

Horizontal format: Group like terms and simplifyHorizontal format: Group like terms and simplify(3x(3x44-2x-2x33+5x+5x22)+(7x)+(7x22+9x+9x33-2x)=(3x-2x)=(3x44) + (-2x) + (-2x33+9x+9x33) + (5x) + (5x22+7x+7x22) + (-2x)) + (-2x)

= = 3x 3x44+7x+7x33+12x+12x22-2x-2x

Adding PolynomialsAdding Polynomials

Find the sum.Find the sum. (2a(2a22 + 7) + (7a + 7) + (7a22+4a-3)+4a-3) (9b(9b22 – b + 8) + (4b – b + 8) + (4b22 – b – 3) – b – 3) (3z(3z22 + z – 4) + (2z + z – 4) + (2z22 + 2z – 3) + 2z – 3) (8c(8c22 – 4c +1) + (-3c – 4c +1) + (-3c22 + c + 5) + c + 5)

Subtracting PolynomialsSubtracting Polynomials

Find the difference.Find the difference.

(3x(3x22 – 9x) – (2x – 9x) – (2x22 – 5x + 6) – 5x + 6)

Vertical format:Vertical format:

3x3x22 – 9x – 9x 3x 3x22 – 9x – 9x

- (- (2x2x22 – 5x + 6 – 5x + 6)) - - 2x2x22 + 5x – 6 + 5x – 6 YOU MUSTYOU MUST

xx22 – 4x – 6 – 4x – 6 DISTRIBUTEDISTRIBUTE THE THE

NEGATIVENEGATIVE SIGN!!!!!SIGN!!!!!

Subtracting PolynomialsSubtracting Polynomials Find the difference.Find the difference.

(3x(3x22 – 9x) – (2x – 9x) – (2x22 – 5x + 6) – 5x + 6)

Horizontal formatHorizontal format: : DISTRIBUTE THE NEGATIVE SIGN!!!DISTRIBUTE THE NEGATIVE SIGN!!!

3x3x22 – 9x – 2x – 9x – 2x22 + 5x – 6 + 5x – 6

Group like terms and simplify.Group like terms and simplify.

(3x(3x22-2x-2x22) + (-9x + 5x) + (-6)) + (-9x + 5x) + (-6)= x= x22 – 4x – 6 – 4x – 6

Subtracting PolynomialsSubtracting Polynomials

Find the differenceFind the difference (7c(7c33 – 6c + 4) – (9c – 6c + 4) – (9c33 – 5c – 5c22 – c) – c) (d(d22 – 15d +10) – (-12d – 15d +10) – (-12d22 + 8d – 1) + 8d – 1) (-4m(-4m22 + 3m – 1) – (m + 2) + 3m – 1) – (m + 2) (3m + 4) – (2m(3m + 4) – (2m22 – 6m + 5) – 6m + 5)

Adding Polynomials to Adding Polynomials to Find PerimeterFind Perimeter

Write a polynomial that represents the Write a polynomial that represents the perimeter of the figure.perimeter of the figure.

Perimeter is the sum of all sides. So, Perimeter is the sum of all sides. So,

P= (2x+4) + (2x+4) + (x-5) + P= (2x+4) + (2x+4) + (x-5) + (3x+1)(3x+1)

3x+1

X-5

2x+42x+4

Vertical Method:Vertical Method:2x + 42x + 42x + 42x + 4 x – 5 x – 5

+ + 3x + 13x + 18x + 48x + 4

Horizontal Method:Horizontal Method:

(2x+4) + (2x+4) + (x-5) + (3x+1) = (2x+2x+x+3x) + (4+4-5+1)(2x+4) + (2x+4) + (x-5) + (3x+1) = (2x+2x+x+3x) + (4+4-5+1) = = 8x + 48x + 4

Adding Polynomials to Adding Polynomials to Find PerimeterFind Perimeter