7.4 - Polynomials. polynomial polynomial – a monomial or sum of monomials

7.4 - Polynomials

polynomial

polynomial – a monomial or sum of monomials

polynomial – a monomial or sum of monomials

Recall:

polynomial – a monomial or sum of monomials

Recall: monomial

polynomial – a monomial or sum of monomials

Recall: monomial – a number,

polynomial – a monomial or sum of monomials

Recall: monomial – a number, a variable,

polynomial – a monomial or sum of monomials

Recall: monomial – a number, a variable, or a product of a number and one or more variables

polynomial – a monomial or sum of monomials

Recall: monomial – a number, a variable, or a product of a number and one or more variables

polynomial – a monomial or sum of monomials

Recall: monomial – a number, a variable, or a product of a number and one or more variables

Ex. 1 State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

polynomial – a monomial or sum of monomials

Recall: monomial – a number, a variable, or a product of a number and one or more variables

Ex. 1 State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

a. 2x – 3yz

polynomial – a monomial or sum of monomials

Recall: monomial – a number, a variable, or a product of a number and one or more variables

Ex. 1 State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

a. 2x – 3yzBinomial

polynomial – a monomial or sum of monomials

Recall: monomial – a number, a variable, or a product of a number and one or more variables

Ex. 1 State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

a. 2x – 3yz Binomial b. 8n3 + 5n-2

polynomial – a monomial or sum of monomials

Recall: monomial – a number, a variable, or a product of a number and one or more variables

Ex. 1 State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

a. 2x – 3yz Binomial b. 8n3 + 5n-2

polynomial – a monomial or sum of monomials

Recall: monomial – a number, a variable, or a product of a number and one or more variables

Ex. 1 State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

a. 2x – 3yz Binomial b. 8n3 + 5n-2

Not a polynomial

polynomial – a monomial or sum of monomials

Recall: monomial – a number, a variable, or a product of a number and one or more variables

Ex. 1 State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

a. 2x – 3yz c. -8 Binomial b. 8n3 + 5n-2

Not a polynomial

polynomial – a monomial or sum of monomials

Recall: monomial – a number, a variable, or a product of a number and one or more variables

Ex. 1 State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

a. 2x – 3yz c. -8 Binomial Monomialb. 8n3 + 5n-2

Not a polynomial

polynomial – a monomial or sum of monomials

Recall: monomial – a number, a variable, or a product of a number and one or more variables

Ex. 1 State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

a. 2x – 3yz c. -8 Binomial Monomialb. 8n3 + 5n-2 d. 4a3 + 5a2 + 6a – 9 Not a polynomial

polynomial – a monomial or sum of monomials

Recall: monomial – a number, a variable, or a product of a number and one or more variables

Ex. 1 State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

a. 2x – 3yz c. -8 Binomial Monomialb. 8n3 + 5n-2 d. 4a3 + 5a2 + 6a – 9 Not a polynomial Polynomial

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

3x

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

area shaded region

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

area shaded region

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

area shaded region =

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

area shaded region =

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

area shaded region = area rectangle

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

area shaded region = area rectangle

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

area shaded region = area rectangle –

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

area shaded region = area rectangle –

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

area shaded region = area rectangle – area circle

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

area shaded region = area rectangle – area circle

3. Write an equation.

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

area shaded region = area rectangle – area circle

3. Write an equation.

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

area shaded region = area rectangle – area circle

3. Write an equation.

AS

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

area shaded region = area rectangle – area circle

3. Write an equation.

AS =

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

area shaded region = area rectangle – area circle

3. Write an equation.

AS = AR

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3x

The area of the shaded region is the area of the rectangle minus the area of the circle.

2. Shorten.

area shaded region = area rectangle – area circle

3. Write an equation.

AS = AR

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3xThe area of the shaded region is the area of

the rectangle minus the area of the circle.2. Shorten.

area shaded region = area rectangle – area circle3. Write an equation.

AS = AR – AC

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3xThe area of the shaded region is the area of

the rectangle minus the area of the circle.2. Shorten.

area shaded region = area rectangle – area circle3. Write an equation.

AS = AR – AC

A

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3xThe area of the shaded region is the area of

the rectangle minus the area of the circle.2. Shorten.

area shaded region = area rectangle – area circle3. Write an equation.

AS = AR – AC

A =

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3xThe area of the shaded region is the area of

the rectangle minus the area of the circle.2. Shorten.

area shaded region = area rectangle – area circle3. Write an equation.

AS = AR – AC

A =

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3xThe area of the shaded region is the area of

the rectangle minus the area of the circle.2. Shorten.

area shaded region = area rectangle – area circle3. Write an equation.

AS = AR – AC

A = (lw)

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3xThe area of the shaded region is the area of

the rectangle minus the area of the circle.2. Shorten.

area shaded region = area rectangle – area circle3. Write an equation.

AS = AR – AC

A = (lw)

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3xThe area of the shaded region is the area of

the rectangle minus the area of the circle.2. Shorten.

area shaded region = area rectangle – area circle3. Write an equation.

AS = AS – AS

A = (lw) – πr2

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3xThe area of the shaded region is the area of

the rectangle minus the area of the circle.2. Shorten.

area shaded region = area rectangle – area circle3. Write an equation.

AS = AS – AS

A = (lw) – πr2

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3xThe area of the shaded region is the area of the

rectangle minus the area of the circle.2. Shorten.

area shaded region = area rectangle – area circle3. Write an equation.

AS = AS – AS

A = (lw) – πr2

A = (2x)(3x)

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3xThe area of the shaded region is the area of the

rectangle minus the area of the circle.2. Shorten.

area shaded region = area rectangle – area circle3. Write an equation.

AS = AS – AS

A = (lw) – πr2

A = (2x)(3x)

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3xThe area of the shaded region is the area of the

rectangle minus the area of the circle.2. Shorten.

area shaded region = area rectangle – area circle3. Write an equation.

AS = AS – AS

A = (lw) – πr2

A = (2x)(3x) – (3.14)(x)2

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3xThe area of the shaded region is the area of the rectangle

minus the area of the circle.2. Shorten.

area shaded region = area rectangle – area circle3. Write an equation.

AS = AS – AS

A = (lw) – πr2

A = (2x)(3x) – (3.14)(x)2

A = 6x2 – 3.14x2

x

Ex. 2 Write a polynomial to represent the area of the shaded region.

2x

1. Write in words. 3xThe area of the shaded region is the area of the rectangle

minus the area of the circle.2. Shorten.

area shaded region = area rectangle – area circle3. Write an equation.

AS = AS – AS

A = (lw) – πr2

A = (2x)(3x) – (3.14)(x)2

A = 6x2 – 3.14x2

A = 2.86x2

x

degree

degree (of a monomial)

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a monomial) – the sum of the exponents of all its variables

degree

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial)

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

Ex. 3 Find the degree of each polynomial.

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

Ex. 3 Find the degree of each polynomial.

a. 5mn2

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

Ex. 3 Find the degree of each polynomial.

a. 5mn2

degree of polynomial = 3

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

Ex. 3 Find the degree of each polynomial.a. 5mn2

degree of polynomial = 3b. -4x2y2 + 3x2 + 5

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

Ex. 3 Find the degree of each polynomial.a. 5mn2

degree of polynomial = 3b. -4x2y2 + 3x2 + 5

(degrees)

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

Ex. 3 Find the degree of each polynomial.a. 5mn2

degree of polynomial = 3b. -4x2y2 + 3x2 + 5

(degrees) 4

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

Ex. 3 Find the degree of each polynomial.a. 5mn2

degree of polynomial = 3b. -4x2y2 + 3x2 + 5

(degrees) 4 2

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

Ex. 3 Find the degree of each polynomial.

a. 5mn2

degree of polynomial = 3

b. -4x2y2 + 3x2 + 5

(degrees) 4 2 0

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

Ex. 3 Find the degree of each polynomial.a. 5mn2

degree of polynomial = 3b. -4x2y2 + 3x2 + 5

(degrees) 4 2 0degree of polynomial = 4

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

Ex. 3 Find the degree of each polynomial.a. 5mn2

degree of polynomial = 3b. -4x2y2 + 3x2 + 5

(degrees) 4 2 0degree of polynomial = 4

c. 3a + 7ab – 2a2b + 16

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

Ex. 3 Find the degree of each polynomial.a. 5mn2

degree of polynomial = 3b. -4x2y2 + 3x2 + 5

(degrees) 4 2 0degree of polynomial = 4

c. 3a + 7ab – 2a2b + 16(degrees)

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

Ex. 3 Find the degree of each polynomial.a. 5mn2

degree of polynomial = 3b. -4x2y2 + 3x2 + 5

(degrees) 4 2 0degree of polynomial = 4

c. 3a + 7ab – 2a2b + 16(degrees) 1

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

Ex. 3 Find the degree of each polynomial.a. 5mn2

degree of polynomial = 3b. -4x2y2 + 3x2 + 5

(degrees) 4 2 0degree of polynomial = 4

c. 3a + 7ab – 2a2b + 16(degrees) 1 2

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

Ex. 3 Find the degree of each polynomial.a. 5mn2

degree of polynomial = 3b. -4x2y2 + 3x2 + 5

(degrees) 4 2 0degree of polynomial = 4

c. 3a + 7ab – 2a2b + 16(degrees) 1 2 3

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

Ex. 3 Find the degree of each polynomial.a. 5mn2

degree of polynomial = 3b. -4x2y2 + 3x2 + 5

(degrees) 4 2 0degree of polynomial = 4

c. 3a + 7ab – 2a2b + 16(degrees) 1 2 3 0

degree (of a monomial) – the sum of the exponents of all its variables

degree (of a polynomial) – the highest degree of a single term in the polynomial

Ex. 3 Find the degree of each polynomial.a. 5mn2

degree of polynomial = 3b. -4x2y2 + 3x2 + 5

(degrees) 4 2 0degree of polynomial = 4

c. 3a + 7ab – 2a2b + 16(degrees) 1 2 3 0

degree of polynomial = 3

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

b. 2xy3 + y2 + 5x3 – 3x2y

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

b. 2xy3 + y2 + 5x3 – 3x2y

y2

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

b. 2xy3 + y2 + 5x3 – 3x2y

y2 + 2xy3

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

b. 2xy3 + y2 + 5x3 – 3x2y

y2 + 2xy3 – 3x2y

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

b. 2xy3 + y2 + 5x3 – 3x2y

y2 + 2xy3 – 3x2y + 5x3

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

b. 2xy3 + y2 + 5x3 – 3x2y y2 + 2xy3 – 3x2y + 5x3

Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order.

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

b. 2xy3 + y2 + 5x3 – 3x2y

y2 + 2xy3 – 3x2y + 5x3

Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order.

a. 6x2 + 5 – 8x – 2x3

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

b. 2xy3 + y2 + 5x3 – 3x2y

y2 + 2xy3 – 3x2y + 5x3

Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order.

a. 6x2 + 5 – 8x – 2x3

-2x3

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

b. 2xy3 + y2 + 5x3 – 3x2y

y2 + 2xy3 – 3x2y + 5x3

Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order.

a. 6x2 + 5 – 8x – 2x3

-2x3 + 6x2

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

b. 2xy3 + y2 + 5x3 – 3x2y

y2 + 2xy3 – 3x2y + 5x3

Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order.

a. 6x2 + 5 – 8x – 2x3

-2x3 + 6x2 – 8x

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

b. 2xy3 + y2 + 5x3 – 3x2y

y2 + 2xy3 – 3x2y + 5x3

Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order.

a. 6x2 + 5 – 8x – 2x3

-2x3 + 6x2 – 8x + 5

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

b. 2xy3 + y2 + 5x3 – 3x2y y2 + 2xy3 – 3x2y + 5x3

Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order.

a. 6x2 + 5 – 8x – 2x3

-2x3 + 6x2 – 8x + 5

b. 3a3x2 – a4 + 4ax5 + 9a2x

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

b. 2xy3 + y2 + 5x3 – 3x2y y2 + 2xy3 – 3x2y + 5x3

Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order.

a. 6x2 + 5 – 8x – 2x3

-2x3 + 6x2 – 8x + 5

b. 3a3x2 – a4 + 4ax5 + 9a2x 4ax5

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

b. 2xy3 + y2 + 5x3 – 3x2y y2 + 2xy3 – 3x2y + 5x3

Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order.

a. 6x2 + 5 – 8x – 2x3

-2x3 + 6x2 – 8x + 5

b. 3a3x2 – a4 + 4ax5 + 9a2x 4ax5 + 3a3x2

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

b. 2xy3 + y2 + 5x3 – 3x2y y2 + 2xy3 – 3x2y + 5x3

Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order.

a. 6x2 + 5 – 8x – 2x3

-2x3 + 6x2 – 8x + 5

b. 3a3x2 – a4 + 4ax5 + 9a2x 4ax5 + 3a3x2 + 9a2x

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order.

a. 7x2 + 2x4 – 11 - 11 + 7x2 + 2x4

b. 2xy3 + y2 + 5x3 – 3x2y y2 + 2xy3 – 3x2y + 5x3

Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order.

a. 6x2 + 5 – 8x – 2x3

-2x3 + 6x2 – 8x + 5

b. 3a3x2 – a4 + 4ax5 + 9a2x 4ax5 + 3a3x2 + 9a2x – a4