7.4 - polynomials. polynomial polynomial – a monomial or sum of monomials
TRANSCRIPT
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