GR 8 Math Powerpoint about Polynomial Techniquesreginaatin
-This is a powerpoint inspired by one of Canva displayed presentation.
- This is about Math Polynomials and good for highschoolers presentation for school.
- It consists of 39 pages explaining each of the Polynomial Techniques.
- Good for review or inspired powerpoint.
GR 8 Math Powerpoint about Polynomial Techniquesreginaatin
-This is a powerpoint inspired by one of Canva displayed presentation.
- This is about Math Polynomials and good for highschoolers presentation for school.
- It consists of 39 pages explaining each of the Polynomial Techniques.
- Good for review or inspired powerpoint.
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
Un grupo de variables representadas por letras junto con un conjunto de números combinados con operaciones de suma, resta, multiplicación, división, potencia o extracción de raíces es llamado una expresión algebraica. Las expresiones algebraicas nos permiten, por ejemplo, hallar áreas y volúmenes
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
2. A mathematics expression is a calculation procedure which
is written using numbers, variables, and operation-symbols.
Polynomial Expressions
3. A mathematics expression is a calculation procedure which
is written using numbers, variables, and operation-symbols.
The purposes of this symbolic-form are clarity and simplicity.
Polynomial Expressions
4. For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
A mathematics expression is a calculation procedure which
is written using numbers, variables, and operation-symbols.
The purposes of this symbolic-form are clarity and simplicity.
Polynomial Expressions
5. For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
A mathematics expression is a calculation procedure which
is written using numbers, variables, and operation-symbols.
The purposes of this symbolic-form are clarity and simplicity.
Polynomial Expressions
The English phrase
“sum 2 and 3 times x”
is ambiguous.
6. For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times x from 4 times the square of x”,
A mathematics expression is a calculation procedure which
is written using numbers, variables, and operation-symbols.
The purposes of this symbolic-form are clarity and simplicity.
Polynomial Expressions
The English phrase
“sum 2 and 3 times x”
is ambiguous.
7. For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times x from 4 times the square of x”,
(3 – 2x)2 means to
“square of the difference of 3 and twice x”.
A mathematics expression is a calculation procedure which
is written using numbers, variables, and operation-symbols.
The purposes of this symbolic-form are clarity and simplicity.
Polynomial Expressions
The English phrase
“sum 2 and 3 times x”
is ambiguous.
8. For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times x from 4 times the square of x”,
(3 – 2x)2 means to
“square of the difference of 3 and twice x”.
A mathematics expression is a calculation procedure which
is written using numbers, variables, and operation-symbols.
The purposes of this symbolic-form are clarity and simplicity.
Polynomial Expressions
The English phrase
“sum 2 and 3 times x”
is ambiguous.
These are complicated!
9. For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times x from 4 times the square of x”,
(3 – 2x)2 means to
“square of the difference of 3 and twice x”.
A mathematics expression is a calculation procedure which
is written using numbers, variables, and operation-symbols.
The purposes of this symbolic-form are clarity and simplicity.
The simplest form of expression are #xN, where N is a non-
negative integer and # is a number, is called a monomial
(one-term).
Polynomial Expressions
The English phrase
“sum 2 and 3 times x”
is ambiguous.
These are complicated!
10. For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times x from 4 times the square of x”,
(3 – 2x)2 means to
“square of the difference of 3 and twice x”.
A mathematics expression is a calculation procedure which
is written using numbers, variables, and operation-symbols.
The purposes of this symbolic-form are clarity and simplicity.
The simplest form of expression are #xN, where N is a non-
negative integer and # is a number, is called a monomial
(one-term). For example –1, 2x, 3x2, and –4x3 are monomials.
Polynomial Expressions
The English phrase
“sum 2 and 3 times x”
is ambiguous.
These are complicated!
11. For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times x from 4 times the square of x”,
(3 – 2x)2 means to
“square of the difference of 3 and twice x”.
A mathematics expression is a calculation procedure which
is written using numbers, variables, and operation-symbols.
The purposes of this symbolic-form are clarity and simplicity.
The simplest form of expression are #xN, where N is a non-
negative integer and # is a number, is called a monomial
(one-term). For example –1, 2x, 3x2, and –4x3 are monomials.
If N = 0 we’ve the constants, N = 1, the linear monomials #x.
Polynomial Expressions
The English phrase
“sum 2 and 3 times x”
is ambiguous.
These are complicated!
12. For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times x from 4 times the square of x”,
(3 – 2x)2 means to
“square of the difference of 3 and twice x”.
A mathematics expression is a calculation procedure which
is written using numbers, variables, and operation-symbols.
The purposes of this symbolic-form are clarity and simplicity.
Example A. Evaluate the monomials if y = –4
a. 3y2
The simplest form of expression are #xN, where N is a non-
negative integer and # is a number, is called a monomial
(one-term). For example –1, 2x, 3x2, and –4x3 are monomials.
If N = 0 we’ve the constants, N = 1, the linear monomials #x.
Polynomial Expressions
The English phrase
“sum 2 and 3 times x”
is ambiguous.
These are complicated!
13. For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times x from 4 times the square of x”,
(3 – 2x)2 means to
“square of the difference of 3 and twice x”.
A mathematics expression is a calculation procedure which
is written using numbers, variables, and operation-symbols.
The purposes of this symbolic-form are clarity and simplicity.
Example A. Evaluate the monomials if y = –4
a. 3y2
3y2 3(–4)2
The simplest form of expression are #xN, where N is a non-
negative integer and # is a number, is called a monomial
(one-term). For example –1, 2x, 3x2, and –4x3 are monomials.
If N = 0 we’ve the constants, N = 1, the linear monomials #x.
Polynomial Expressions
The English phrase
“sum 2 and 3 times x”
is ambiguous.
These are complicated!
14. For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times x from 4 times the square of x”,
(3 – 2x)2 means to
“square of the difference of 3 and twice x”.
A mathematics expression is a calculation procedure which
is written using numbers, variables, and operation-symbols.
The purposes of this symbolic-form are clarity and simplicity.
Example A. Evaluate the monomials if y = –4
a. 3y2
3y2 3(–4)2 = 3(16) = 48
The simplest form of expression are #xN, where N is a non-
negative integer and # is a number, is called a monomial
(one-term). For example –1, 2x, 3x2, and –4x3 are monomials.
If N = 0 we’ve the constants, N = 1, the linear monomials #x.
Polynomial Expressions
The English phrase
“sum 2 and 3 times x”
is ambiguous.
These are complicated!
23. b. –3y2 (y = –4)
–3y2 –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called polynomials (many-terms).
Polynomial Expressions
Polynomial Expressions
24. b. –3y2 (y = –4)
–3y2 –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called polynomials (many-terms).
These are expressions of the form, arranged in the order of
powers of the x: #xN ± #xN-1 ± … ± #x1 ± #
where the #’s are numbers.
Polynomial Expressions
Polynomial Expressions
25. b. –3y2 (y = –4)
–3y2 –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called polynomials (many-terms).
These are expressions of the form, arranged in the order of
powers of the x: #xN ± #xN-1 ± … ± #x1 ± #
where the #’s are numbers.
The highest exponent N is the degree of the polynomial.
Polynomial Expressions
Polynomial Expressions
26. b. –3y2 (y = –4)
–3y2 –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called polynomials (many-terms).
These are expressions of the form, arranged in the order of
powers of the x: #xN ± #xN-1 ± … ± #x1 ± #
where the #’s are numbers.
The highest exponent N is the degree of the polynomial.
For example, 4x – 7 is 1st degree (linear)
Polynomial Expressions
Polynomial Expressions
27. b. –3y2 (y = –4)
–3y2 –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called polynomials (many-terms).
These are expressions of the form, arranged in the order of
powers of the x: #xN ± #xN-1 ± … ± #x1 ± #
where the #’s are numbers.
The highest exponent N is the degree of the polynomial.
For example, 4x – 7 is 1st degree (linear)
and the degree of 1 – 3x2 – πx40 is 40.
Polynomial Expressions
Polynomial Expressions
28. b. –3y2 (y = –4)
–3y2 –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called polynomials (many-terms).
These are expressions of the form, arranged in the order of
powers of the x: #xN ± #xN-1 ± … ± #x1 ± #
where the #’s are numbers.
The highest exponent N is the degree of the polynomial.
For example, 4x – 7 is 1st degree (linear)
and the degree of 1 – 3x2 – πx40 is 40.
x
1 is not a polynomial.The expression
Polynomial Expressions
Polynomial Expressions
29. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
Polynomial Expressions
30. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3.
Polynomial Expressions
31. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Polynomial Expressions
32. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression,
Polynomial Expressions
33. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
Polynomial Expressions
34. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
= 4(9) – 3(–27)
Polynomial Expressions
35. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
= 4(9) – 3(–27)
= 36 + 81
= 117
Polynomial Expressions
36. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
= 4(9) – 3(–27)
= 36 + 81
= 117
Given a polynomial, each monomial is called a term.
Polynomial Expressions
37. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
= 4(9) – 3(–27)
= 36 + 81
= 117
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Polynomial Expressions
38. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 and –3x3. When evaluating the polynomial,
we evaluate each monomial then combine the results.
Set x = (–3) in the expression, we get
4(–3)2 – 3(–3)3
= 4(9) – 3(–27)
= 36 + 81
= 117
Given a polynomial, each monomial is called a term.
#xN ± #xN-1 ± … ± #x ± #
terms
Therefore the polynomial –3x2 – 4x + 7 has 3 terms,
–3x2 , –4x and + 7.
Polynomial Expressions
39. Each term is addressed by the variable part.
Polynomial Expressions
40. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2,
Polynomial Expressions
41. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
Polynomial Expressions
42. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
Polynomial Expressions
43. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term.
Polynomial Expressions
44. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Polynomial Expressions
45. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Operations with Polynomials
Polynomial Expressions
46. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Operations with Polynomials
Polynomial Expressions
47. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
Operations with Polynomials
Polynomial Expressions
48. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x
Operations with Polynomials
Polynomial Expressions
49. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Operations with Polynomials
Polynomial Expressions
50. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined.
Operations with Polynomials
Polynomial Expressions
51. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Operations with Polynomials
Polynomial Expressions
52. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
Operations with Polynomials
Polynomial Expressions
53. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it with the
coefficient.
Operations with Polynomials
Polynomial Expressions
54. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it with the
coefficient. Hence, 3(5x) = (3*5)x
Operations with Polynomials
Polynomial Expressions
55. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it with the
coefficient. Hence, 3(5x) = (3*5)x =15x,
Operations with Polynomials
Polynomial Expressions
56. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it with the
coefficient. Hence, 3(5x) = (3*5)x =15x,
and –2(–4x) = (–2)(–4)x = 8x.
Operations with Polynomials
Polynomial Expressions
57. Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the number term or the constant term is 7.
The number in front of a term is called the coefficient of that
term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms.
Like-terms may be combined.
For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Unlike terms may not be combined. So x + x2 stays as x + x2.
Note that we write 1xN as xN , –1xN as –xN.
When multiplying a number with a term, we multiply it with the
coefficient. Hence, 3(5x) = (3*5)x =15x,
and –2(–4x) = (–2)(–4)x = 8x.
Operations with Polynomials
When multiplying a number with a polynomial, we may
expand using the distributive law: A(B ± C) = AB ± AC.
Polynomial Expressions
58. Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
Polynomial Expressions
59. Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
Polynomial Expressions
60. Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
Polynomial Expressions
61. Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
Polynomial Expressions
64. Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomials in two or more variables.
Polynomial Expressions
65. Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers
Polynomial Expressions
66. Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2
Polynomial Expressions
67. Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Polynomial Expressions
68. Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
Polynomial Expressions
69. Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3
Polynomial Expressions
70. Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined.
Polynomial Expressions
71. Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined. We evaluate them by assigning numbers to
x and/or y.
Polynomial Expressions
72. Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined. We evaluate them by assigning numbers to
x and/or y. If only one number is given, the result is a formula.
Polynomial Expressions
73. Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined. We evaluate them by assigning numbers to
x and/or y. If only one number is given, the result is a formula.
If both numbers are given, then we get a numerical output.
Polynomial Expressions
74. Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
= –3x2 + 9x – 15 + 2x2 + 8x +12
= –x2 + 17x – 3
Polynomials in two or more variables.
We form polynomials in two variables say, x & y, by adding
monomials of the form kx#y# where k is a number and the
powers are all nonnegative integers such as –5x3y2 or 3x2.
Like–terms are terms where the variable parts are the same.
For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be
combined. We evaluate them by assigning numbers to
x and/or y. If only one number is given, the result is a formula.
If both numbers are given, then we get a numerical output.
We may do this for x, y and z or even more variables.
Polynomial Expressions
75. Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
Polynomial Expressions
76. = 6xy – 8x2y + 2xy – 3xy2
Polynomial Expressions
Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
77. Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
Polynomial Expressions
78. Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Polynomial Expressions
79. Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2
Polynomial Expressions
80. Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2
= 16y – 32y – 6y2
Polynomial Expressions
81. Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2
= 16y – 32y – 6y2
= –16y – 6y2
Polynomial Expressions
82. Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2
= 16y – 32y – 6y2
= –16y – 6y2
c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3
Polynomial Expressions
83. Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2
= 16y – 32y – 6y2
= –16y – 6y2
c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3
We may put x = 2, y = 3 into the formula and do everything
all over
Polynomial Expressions
84. Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2
= 16y – 32y – 6y2
= –16y – 6y2
c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3
We may put x = 2, y = 3 into the formula and do everything
all over again or we may plug into y = 3 into part b which is
easier.
Polynomial Expressions
85. Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2
= 16y – 32y – 6y2
= –16y – 6y2
c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3
We may put x = 2, y = 3 into the formula and do everything
all over again or we may plug into y = 3 into part b which is
easier. We will do the easy way.
Polynomial Expressions
86. Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2
= 16y – 32y – 6y2
= –16y – 6y2
c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3
We may put x = 2, y = 3 into the formula and do everything
all over again or we may plug into y = 3 into part b which is
easier. We will do the easy way.
Input y = 3 into –16y – 6y2
Polynomial Expressions
87. Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2
= 16y – 32y – 6y2
= –16y – 6y2
c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3
We may put x = 2, y = 3 into the formula and do everything
all over again or we may plug into y = 3 into part b which is
easier. We will do the easy way.
–16(3) – 6(3)2
Input y = 3 into –16y – 6y2
we get
Polynomial Expressions
88. Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.
Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2
= 16y – 32y – 6y2
= –16y – 6y2
c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3
We may put x = 2, y = 3 into the formula and do everything
all over again or we may plug into y = 3 into part b which is
easier. We will do the easy way.
–16(3) – 6(3)2
Input y = 3 into –16y – 6y2
we get
= –48 – 54 = –102
Polynomial Expressions
89. Ex. A. Evaluate each monomials with the given values.
3. 2x2 with x = 1 and x = –1 4. –2x2 with x = 1 and x = –1
5. 5y3 with y = 2 and y = –2 6. –5y3 with y = 2 and y = –2
1. 2x with x = 1 and x = –1 2. –2x with x = 1 and x = –1
7. 5z4 with z = 2 and z = –2 8. –5y4 with z = 2 and z = –2
B. Evaluate each monomials with the given values.
9. 2x2 – 3x + 2 with x = 1 and x = –1
10. –2x2 + 4x – 1 with x = 2 and x = –2
11. 3x2 – x – 2 with x = 3 and x = –3
12. –3x2 – x + 2 with x = 3 and x = –3
13. –2x3 – x2 + 4 with x = 2 and x = –2
14. –2x3 – 5x2 – 5 with x = 3 and x = –3
C. Expand and simplify.
15. 5(2x – 4) + 3(4 – 5x) 16. 5(2x – 4) – 3(4 – 5x)
17. –2(3x – 8) + 3(4 – 9x) 18. –2(3x – 8) – 3(4 – 9x)
19. 7(–2x – 7) – 3(4 – 3x) 20. –5(–2 – 8x) + 7(–2 – 11x)
Polynomial Expressions
90. 21. x2 – 3x + 5 + 2(–x2 – 4x – 6)
22. x2 – 3x + 5 – 2(–x2 – 4x – 6)
23. 2(x2 – 3x + 5) + 5(–x2 – 4x – 6)
24. 2(x2 – 3x + 5) – 5(–x2 – 4x – 6)
25. –2(3x2 – 2x + 5) + 5(–4x2 – 4x – 3)
26. –2(3x2 – 2x + 5) – 5(–4x2 – 4x – 3)
27. 4(3x3 – 5x2) – 9(6x2 – 7x) – 5(– 8x – 2)
29. Simplify 2(3xy – xy2) – 2(2xy – xy2) then evaluated it
with x = –1, afterwards evaluate it at (–1, 2) for (x, y)
30. Simplify x2 – 2(3xy – x2) – 2(y2 – xy) then evaluated it
with y = –2, afterwards evaluate it at (–1, –2) for (x, y)
31. Simplify x2 – 2(3xy – z2) – 2(z2 – x2) then evaluated it
with x = –1, y = – 2 and z = 3.
Polynomial Expressions
28. –6(7x2 + 5x – 9) – 7(–3x2 – 2x – 7)