The document discusses techniques for factoring polynomials. It explains how to factor the difference and sum of two squares, perfect square trinomials, and the sum and difference of two cubes. For each type of factorization, it provides steps to follow, such as taking the square root of terms for differences of squares or cube roots for sums and differences of cubes. Examples are worked through applying the steps to factor various polynomials.
This will help you in factoring sum and difference of two cubes.
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This will help you in factoring sum and difference of two cubes.
For more instructional resources, CLICK me here!
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Polynomials are algebraic expressions that are consist of variables and coefficients. We can perform arithmetic operations such as subtraction, addition, multiplication and division. This presentation is all about factoring completely different types of polynomials. There four types of polynomials to factor that would be discuss in this presentation
If you are looking for math video tutorials (with voice recording), you may download it on our YouTube Channel. Don't forget to SUBSCRIBE for you to get updated on our upcoming videos.
https://tinyurl.com/y9muob6q
Also, please do visit our page, LIKE and FOLLOW us on Facebook!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
Polynomials are algebraic expressions that are consist of variables and coefficients. We can perform arithmetic operations such as subtraction, addition, multiplication and division. This presentation is all about factoring completely different types of polynomials. There four types of polynomials to factor that would be discuss in this presentation
If you are looking for math video tutorials (with voice recording), you may download it on our YouTube Channel. Don't forget to SUBSCRIBE for you to get updated on our upcoming videos.
https://tinyurl.com/y9muob6q
Also, please do visit our page, LIKE and FOLLOW us on Facebook!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
Strength of Materials all formulas in pdf
it subject iosd also klnown as mechanics of Soilid.
in this pdf there are formulas of stress strain springs - closed coil helical spring , open coil helical Springs etc.
You already know relationships where one variable varies directly or inversely with another variable.
Now you will look at relationships where one variable varies directly with two or more other variables but does not vary inversely with any other variable.
All around us, some quantities are constant and others are variable.
For instance, the number of hours in a day is constant, but the number of hours in a daylight in a day is not.
now you will explore more closely certain types of relationships between variables.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
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.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
3. Factoring
the
Difference
of Two
Squares
The difference of two squares a2
and b2 has factors with the same
first and last terms.
a2 – b2 = (a + b)(a – b)
Take note that this form of
factoring only works when the
first and last terms of the given
binomial are perfect squares and
the operation between them is
subtraction.
5. Example: Factor the following completely.
1. x2 – y2
Solution:
1. Square root the first term.
2. Square root the last term.
3. Multiply the sum and
difference of the first and
last term.
𝑥2 = 𝑥
𝑦2 = 𝑦
𝒙 + 𝒚 𝒙 − 𝒚
= 𝒙 + 𝒚)(𝒙 − 𝒚
6. Example: Factor the following completely.
2. 𝟒𝒘 𝟐
− 𝟐𝟓
Solution:
1. Square root the first term.
2. Square root the last term.
3. Multiply the sum and
difference of the first and
last term.
𝟒𝐰2= 𝟐𝐰
𝟐𝟓 = 𝟓
𝟐𝐰 + 𝟓 𝟐𝐰 − 𝟓
= 𝟐𝐰 + 𝟓)(𝟐𝐰 − 𝟓
7. Example: Factor the following completely.
3. −𝟑𝟔 + 𝒑 𝟒
Solution:
1. Square root the first term.
2. Square root the last term.
3. Multiply the sum and
difference of the first and
last term.
𝒑 𝟒 = 𝐩 𝟐
𝟑𝟔 = 𝟔
𝐩 𝟐
+ 𝟔 𝐩 𝟐
− 𝟔
= 𝒑 𝟐
+ 𝟔)(𝐩 𝟐
− 𝟔
𝒑 𝟒
− 𝟑𝟔
8. Example: Factor the following completely.
4.𝟏𝟔𝒅 𝟐
𝒆 𝟒
− 𝟔𝟒𝒇 𝟔
𝒈 𝟖
Solution:
1. Square root the first term.
2. Square root the last term.
3. Multiply the sum and
difference of the first and last
term.
𝟏𝟔𝒅 𝟐 𝒆 𝟒 = 𝟒𝐝𝐞 𝟐
𝟔𝟒𝒇 𝟔 𝒈 𝟖
= 𝟖𝐟 𝟑
𝐠 𝟒
𝟒𝐝𝐞 𝟐
+ 𝟖𝐟 𝟑
𝐠 𝟒 𝟒𝐝𝐞 𝟐
− 𝟖𝐟 𝟑
𝐠 𝟒
= 𝟒𝐝𝐞 𝟐
+ 𝟖𝐟 𝟑
𝐠 𝟒
)(𝟒𝐝𝐞 𝟐
− 𝟖𝐟 𝟑
𝐠 𝟒
9. Factoring
Perfect
Square
Trinomials
Polynomials of the form
𝒙 𝟐
+ 𝟐𝒙𝒚 + 𝒚 𝟐
𝒂𝒏𝒅
𝒙 𝟐
− 𝟐𝒙𝒚 + 𝒚 𝟐
are perfect square
trinomials. The first and last terms
of these polynomials are perfect
squares, whereas the middle term
is twice the product of the squares
of the first and last term. These
polynomials can be expressed as
products of two binomials:
10. Factoring
Perfect
Square
Trinomials
𝒙 𝟐
+ 𝟐𝒙𝒚 + 𝒚 𝟐
= (𝒙 + 𝒚)(𝒙 + 𝒚)
𝒙 𝟐
− 𝟐𝒙𝒚 + 𝒚 𝟐
= 𝒙 − 𝒚 𝒙 − 𝒚
The first and last terms of the
factors are square roots of the
first and last terms of the
product.
11. To factor the
perfect
square
trinomials:
1. Square root the first term.
2. Square root the last term.
3.The + or – sign of the middle
term of the product will be
carried as the operation
between the terms in the
factor.
12. Example: Factor the following completely.
𝒂. 𝒙 𝟐
− 𝟒𝒙 + 𝟒
Solution:
1. Square root the first term.
2. Square root the last term.
𝒙 𝟐 = 𝒙
𝟒 = 𝟐
𝒙 − 𝟐 𝒙 − 𝟐
= 𝒙 − 𝟐)(𝒙 − 𝟐
3.The + or – sign of the middle term
of the product will be carried as
the operation between the terms
in the factor.
13. Example: Factor the following completely.
𝒃. 𝒎 𝟐
+ 𝟐𝟎𝒎 + 𝟏𝟎𝟎
Solution:
1. Square root the first term.
2. Square root the last term.
𝒎 𝟐 = 𝒎
𝟏𝟎𝟎 = 𝟏𝟎
𝒎 + 𝟏𝟎 𝒎 + 𝟏𝟎
= 𝒎 + 𝟏𝟎)(𝒎 + 𝟏𝟎
3.The + or – sign of the middle term
of the product will be carried as
the operation between the terms
in the factor.
14. Example: Factor the following completely.
𝒄. 𝒏 𝟐
− 𝟏𝟔𝒏 + 𝟔𝟒
Solution:
1. Square root the first term.
2. Square root the last term.
𝒏 𝟐 = 𝐧
𝟔𝟒 = 𝟖
𝒏 − 𝟖 𝒏 − 𝟖
= 𝒏 − 𝟖)(𝒏 − 𝟖
3.The + or – sign of the middle term
of the product will be carried as
the operation between the terms
in the factor.
15. Example: Factor the following completely.
𝒅. 𝟒𝒑 𝟐
+ 𝟐𝟎𝒑 + 𝟐𝟓
Solution:
1. Square root the first term.
2. Square root the last term.
𝟒𝒑 𝟐
= 𝟐𝐩
𝟐𝟓 = 𝟓
𝟐𝒑 + 𝟓 𝟐𝒑 + 𝟓
= 𝟐𝒑 + 𝟓)(𝟐𝒑 + 𝟓
3.The + or – sign of the middle term
of the product will be carried as
the operation between the terms
in the factor.
16. Factoring
the Sum
and
Difference
of Two
Cubes
The sum and difference of
two cubes are written in the
form of 𝒙 𝟑
+𝒚 𝟑
𝒐𝒓 𝒙 𝟑
− 𝒚 𝟑
.
These polynomials have
factors
(𝒙 + 𝒚)(𝒙 𝟐
− 𝒙𝒚 + 𝒚 𝟐
) 𝒂𝒏𝒅
(𝒙 − 𝒚)(𝒙 𝟐
+ 𝒙𝒚 + 𝒚 𝟐
),
respectively.
17. To factor
the sum and
difference of
two cubes:
1. Cube root the first term.
2. Cube root the last term.
3. “Write What You See”
4. “Square-Multiply-Square”
5. “Same-Different-End on a
Positive”
6. Write the answer.
)𝒙3
± 𝒚3
= (𝒙 ± 𝒚)(𝒙2
∓ 𝒙𝒚 + 𝒚2
18. Example: Factor the following completely.
𝟏. 𝟐𝟕𝒖 𝟑
− 𝟏
Solution:
3
𝟐𝟕𝒖 𝟑 = 𝟑𝐮
3
𝟏 = 𝟏
𝟑𝒖
= (𝟑𝒖 − 𝟏)(𝟗𝒖 𝟐
+ 𝟑𝒖 + 𝟏)
1. Cube root the first term.
2. Cube root the last term.
3. “Write What You See”
4. “Square-Multiply-Square”
5. “Same-Different-End on a Positive”
6. Write the answer.
𝟏
𝟑𝒖𝟗𝒖 𝟐 𝟏
+− +
(𝟑𝒖− 𝟏) (𝟗𝒖 𝟐
+ 𝟑𝒖 + 𝟏)
19. Example: Factor the following completely.
𝟐. 𝒚 𝟏𝟐
+ 𝒛 𝟔
Solution:
3
𝒚 𝟏𝟐
= 𝐲 𝟒
3
𝒛 𝟔 = 𝐳 𝟐
𝒚 𝟒
= (𝒚 𝟒
+ 𝒛 𝟐
)(𝒚 𝟖
− 𝒚 𝟒
𝒛 𝟐
+ 𝒛 𝟒
)
1. Cube root the first term.
2. Cube root the last term.
3. “Write What You See”
4. “Square-Multiply-Square”
5. “Same-Different-End on a Positive”
6. Write the answer.
𝒛 𝟐
𝒚 𝟒
𝒛 𝟐𝒚 𝟖 𝒛 𝟒
−+ +
(𝒚 𝟒 +𝒛 𝟐
) (𝒚 𝟖
− 𝒚 𝟒
𝒛 𝟐 +𝒛 𝟐
)
20. Example: Factor the following completely.
𝟑. 𝟔𝟒 + 𝒗 𝟏𝟓
Solution:
3
𝟔𝟒 = 𝟒
3
𝒗 𝟏𝟓 = 𝒗 𝟓
𝟒
= (𝟒 + 𝒗 𝟓
)(𝟏𝟔 − 𝟒𝒗 𝟓
+ 𝒗 𝟏𝟎
)
1. Cube root the first term.
2. Cube root the last term.
3. “Write What You See”
4. “Square-Multiply-Square”
5. “Same-Different-End on a Positive”
6. Write the answer.
𝒗 𝟓
𝟒𝒗 𝟓𝟏𝟔 𝒗 𝟏𝟎
−+ +
(𝟒 + 𝒗 𝟓
) (𝟏𝟔 − 𝟒𝒗 𝟓 + 𝒗 𝟏𝟎
)
21. Example: Factor the following completely.
𝟒. 𝟖𝒙 𝟑
− 𝟏
Solution:
3
𝟖𝒙 𝟑 = 𝟐𝐱
3
𝟏 = 𝟏
𝟐𝒙
= (𝟐𝒙 − 𝟏)(𝟒𝒙 𝟐
+ 𝟐𝒙 + 𝟏)
1. Cube root the first term.
2. Cube root the last term.
3. “Write What You See”
4. “Square-Multiply-Square”
5. “Same-Different-End on a Positive”
6. Write the answer.
𝟏
𝟐𝒙𝟒𝒙 𝟐 𝟏
+− +
(𝟐𝒙− 𝟏) (𝟒𝒙 𝟐
+ 𝟐𝒙 + 𝟏)
