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 – 3yz Binomial

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. 8n 3 + 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. 8n 3 + 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. 8n 3 + 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. 8n 3 + 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 Monomial b. 8n 3 + 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 Monomial b. 8n 3 + 5n -2 d. 4a 3 + 5a 2 + 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 Monomial b. 8n 3 + 5n -2 d. 4a 3 + 5a 2 + 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. A S 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. A S = 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. A S = A R 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. A S = A R 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. A S = A R – A C 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. A S = A R – A C A 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. A S = A R – A C A = 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. A S = A R – A C A = 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. A S = A R – A C A = (lw) 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. A S = A R – A C A = (lw) 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. A S = A S – A S A = (lw) – πr 2 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. A S = A S – A S A = (lw) – πr 2 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. A S = A S – A S A = (lw) – πr 2 A = (2x)(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. 2. Shorten. area shaded region = area rectangle – area circle 3. Write an equation. A S = A S – A S A = (lw) – πr 2 A = (2x)(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. 2. Shorten. area shaded region = area rectangle – area circle 3. Write an equation. A S = A S – A S A = (lw) – πr 2 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. 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. A S = A S – A S A = (lw) – πr 2 A = (2x)(3x) – (3.14)(x) 2 A = 6x 2 – 3.14x 2 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. A S = A S – A S A = (lw) – πr 2 A = (2x)(3x) – (3.14)(x) 2 A = 6x 2 – 3.14x 2 A = 2.86x 2 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. 5mn 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. 5mn 2 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. 5mn 2 degree of polynomial = 3 b. -4x 2 y 2 + 3x 2 + 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. 5mn 2 degree of polynomial = 3 b. -4x 2 y 2 + 3x (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. 5mn 2 degree of polynomial = 3 b. -4x 2 y 2 + 3x (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. 5mn 2 degree of polynomial = 3 b. -4x 2 y 2 + 3x (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. 5mn 2 degree of polynomial = 3 b. -4x 2 y 2 + 3x (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. 5mn 2 degree of polynomial = 3 b. -4x 2 y 2 + 3x (degrees) degree 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. 5mn 2 degree of polynomial = 3 b. -4x 2 y 2 + 3x (degrees) degree of polynomial = 4 c. 3a + 7ab – 2a 2 b + 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. 5mn 2 degree of polynomial = 3 b. -4x 2 y 2 + 3x (degrees) degree of polynomial = 4 c. 3a + 7ab – 2a 2 b + 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. 5mn 2 degree of polynomial = 3 b. -4x 2 y 2 + 3x (degrees) degree of polynomial = 4 c. 3a + 7ab – 2a 2 b + 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. 5mn 2 degree of polynomial = 3 b. -4x 2 y 2 + 3x (degrees) degree of polynomial = 4 c. 3a + 7ab – 2a 2 b + 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. 5mn 2 degree of polynomial = 3 b. -4x 2 y 2 + 3x (degrees) degree of polynomial = 4 c. 3a + 7ab – 2a 2 b + 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. 5mn 2 degree of polynomial = 3 b. -4x 2 y 2 + 3x (degrees) degree of polynomial = 4 c. 3a + 7ab – 2a 2 b + 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. 5mn 2 degree of polynomial = 3 b. -4x 2 y 2 + 3x (degrees) degree of polynomial = 4 c. 3a + 7ab – 2a 2 b + 16 (degrees) degree of polynomial = 3

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

a. 7x 2 + 2x 4 – 11

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 –

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2 + 2x 4

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2 + 2x 4 b. 2xy 3 + y 2 + 5x 3 – 3x 2 y

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2 + 2x 4 b. 2xy 3 + y 2 + 5x 3 – 3x 2 y y 2

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2 + 2x 4 b. 2xy 3 + y 2 + 5x 3 – 3x 2 y y 2 + 2xy 3

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2 + 2x 4 b. 2xy 3 + y 2 + 5x 3 – 3x 2 y y 2 + 2xy 3 – 3x 2 y

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2 + 2x 4 b. 2xy 3 + y 2 + 5x 3 – 3x 2 y y 2 + 2xy 3 – 3x 2 y + 5x 3

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2 + 2x 4 b. 2xy 3 + y 2 + 5x 3 – 3x 2 y y 2 + 2xy 3 – 3x 2 y + 5x 3 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. 7x 2 + 2x 4 – x 2 + 2x 4 b. 2xy 3 + y 2 + 5x 3 – 3x 2 y y 2 + 2xy 3 – 3x 2 y + 5x 3 Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order. a. 6x – 8x – 2x 3

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2 + 2x 4 b. 2xy 3 + y 2 + 5x 3 – 3x 2 y y 2 + 2xy 3 – 3x 2 y + 5x 3 Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order. a. 6x – 8x – 2x 3 -2x 3

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2 + 2x 4 b. 2xy 3 + y 2 + 5x 3 – 3x 2 y y 2 + 2xy 3 – 3x 2 y + 5x 3 Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order. a. 6x – 8x – 2x 3 -2x 3 + 6x 2

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2 + 2x 4 b. 2xy 3 + y 2 + 5x 3 – 3x 2 y y 2 + 2xy 3 – 3x 2 y + 5x 3 Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order. a. 6x – 8x – 2x 3 -2x 3 + 6x 2 – 8x

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2 + 2x 4 b. 2xy 3 + y 2 + 5x 3 – 3x 2 y y 2 + 2xy 3 – 3x 2 y + 5x 3 Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order. a. 6x – 8x – 2x 3 -2x 3 + 6x 2 – 8x + 5

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2 + 2x 4 b. 2xy 3 + y 2 + 5x 3 – 3x 2 y y 2 + 2xy 3 – 3x 2 y + 5x 3 Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order. a. 6x – 8x – 2x 3 -2x 3 + 6x 2 – 8x + 5 b. 3a 3 x 2 – a 4 + 4ax 5 + 9a 2 x

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2 + 2x 4 b. 2xy 3 + y 2 + 5x 3 – 3x 2 y y 2 + 2xy 3 – 3x 2 y + 5x 3 Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order. a. 6x – 8x – 2x 3 -2x 3 + 6x 2 – 8x + 5 b. 3a 3 x 2 – a 4 + 4ax 5 + 9a 2 x 4ax 5

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2 + 2x 4 b. 2xy 3 + y 2 + 5x 3 – 3x 2 y y 2 + 2xy 3 – 3x 2 y + 5x 3 Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order. a. 6x – 8x – 2x 3 -2x 3 + 6x 2 – 8x + 5 b. 3a 3 x 2 – a 4 + 4ax 5 + 9a 2 x 4ax 5 + 3a 3 x 2

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2 + 2x 4 b. 2xy 3 + y 2 + 5x 3 – 3x 2 y y 2 + 2xy 3 – 3x 2 y + 5x 3 Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order. a. 6x – 8x – 2x 3 -2x 3 + 6x 2 – 8x + 5 b. 3a 3 x 2 – a 4 + 4ax 5 + 9a 2 x 4ax 5 + 3a 3 x 2 + 9a 2 x

Ex. 4 Arrange the terms of each polynomial so that the powers of x are in ascending order. a. 7x 2 + 2x 4 – x 2 + 2x 4 b. 2xy 3 + y 2 + 5x 3 – 3x 2 y y 2 + 2xy 3 – 3x 2 y + 5x 3 Ex. 5 Arrange the terms of each polynomial so that the powers of x are in descending order. a. 6x – 8x – 2x 3 -2x 3 + 6x 2 – 8x + 5 b. 3a 3 x 2 – a 4 + 4ax 5 + 9a 2 x 4ax 5 + 3a 3 x 2 + 9a 2 x – a 4