This document provides instructions for factoring trinomials with leading coefficients of 1 or greater than 1. For trinomials with a leading coefficient of 1, the document explains how to list the factors of the last term, identify the factor pair that sums to the middle term, and write the factors. For trinomials with a leading coefficient greater than 1, the instructions are to find the product of the leading and last terms, identify factor pairs that sum to the middle term, rewrite the trinomial, group terms, and factor. Examples are provided to demonstrate the process.
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.
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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!
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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.
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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
You will learn how to factor polynomials with common monomial factor.
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This powerpoint presentation discusses or talks about the topic or lesson Functions. It also discusses and explains the rules, steps and examples of Quadratic Functions.
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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
You will learn how to factor polynomials with common monomial factor.
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This powerpoint presentation discusses or talks about the topic or lesson Functions. It also discusses and explains the rules, steps and examples of Quadratic Functions.
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.
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Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
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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.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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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”.
1. Prepared by: Gauben L. Malicsi
To factor trinomials with 1 as the numerical coefficient of the leading
term:
a. Factor the leading term of the trinomial and write these factors
as the leading terms of the factors;
b. List down all the factors of the last term;
c. Identify which factor pair sums up to the middle term; then
d. Write each factor in the pairs as the last term of the binomial
factors.
To factor trinomials with more than 1 as the numerical coefficient of
the leading term:
a. Find the product of the leading term and the last term.
b. Find the factors of the product whose sum is equal to the middle
term
c. Rewrite the trinomial as a four-term expression by replacing the
middle term with the sum of the factors.
d. Group terms with common factors.
e. Factor the groups using greatest common monomial factor.
f. Factor out the common binomial factor and write the remaining
factor as a sum or difference of the common monomial factors.
EXAMPLE 6:
Factoring general trinomial with leading coefficient 1
Factor each polynomial.
a. 𝑝2
+ 5𝑝 + 6 b. 𝑥2
+ 8𝑥 − 9
SOLUTIONS:
a. List all the possible factors
of 6.
Factors of 6
2 and 3 -2 and -3
6 and 1 -6 and -1
Find factors of 6 whose sum is 5.
2 + 3 = 5 (-2) + (-3) = -5
6 + 1 = 7 (-6) + (-1) = -7
Thus, the factor of 𝑝2
+ 5𝑝 + 6 =
(𝒑 + 𝟐)(𝒑 + 𝟑)
b. List all the possible factors
of -9.
Factors of -9
-3 and 3 -9 and 1
3 and -3 9 and -1
Find factors of -9 whose sum is 8.
(-3) + 3 = 0 (-9) + 1 = -8
3 + (-3) = 0 9 + (-1) = 8
Thus, the factor of 𝑥2
+ 8𝑥 − 9 =
( 𝒙 + 𝟗)(𝒙 − 𝟕)
EXAMPLE 7:
Factoring general trinomial with leading coefficient greater than 1
Factor each polynomial.
a. 6𝑥2
− 5𝑥 − 6 b. 2𝑦2
− 11𝑦 + 12
SOLUTIONS:
a. 6𝑥2
− 5𝑥 − 6
Find the product of the leading term and the last term.
(6𝑥2
)(−6) = −36𝑥2
Find the factors of −36𝑥2
whose sum is -5x.
(-6x) + 6x = 0 (-9x) + 4x = -5x (-36x) + x = -35x
6x + (-6x) = 0 9x + (-4x) = 5x 36x + (-x) = 35x
Factoring general trinomials
Factors
completely
general
trinomials
COMPETENCY 1.5
2. Prepared by: Gauben L. Malicsi
Rewrite the trinomial as a four-term expression by replacing the
middle term with the sum of the factors.
6𝑥2
− 9𝑥 + 4𝑥 − 6
Group terms with common factors.
(6𝑥2
− 9𝑥) + (4𝑥 − 6)
Factor the groups using greatest common monomial factor.
3x(2x – 3) + 2(2x – 3)
Factor out the common binomial factor and write the remaining factor
as a sum or difference of the common monomial factors.
(2x – 3)(3x + 2)
b. 2𝑦2
− 11𝑦 + 12
Find the product of the leading term and the last term.
(2𝑦2
)(12) = 24𝑦2
Find the factors of 24𝑦2
whose sum is -11y.
(-6y) + (-4y) = -10y (-8y) + (-3y) = -11y (-24y) + (-x) = -25y
6y + 4y = 10y 8y + 3y = 11y 24y + y = 25y
Rewrite the trinomial as a four-term expression by replacing the
middle term with the sum of the factors.
2𝑦2
− 8𝑦 − 3𝑦 + 12
Group terms with common factors.
(2𝑦2
− 8𝑦) − (3𝑦 + 12)
Factor the groups using greatest common monomial factor.
2y(y – 4) - 3(y – 4)
Factor out the common binomial factor and write the remaining factor
as a sum or difference of the common monomial factors.
(y – 4)(2y – 3)
PERFORMANCE TASK:
Direction: Make your own examples of general trinomials including the
factored form.
1 as leading coefficient More than 1 as leading coefficient
Example Factored form Example Factored form
ASSESSMENT TASK:
Show your solutions on the box provided below and write your final
answer on the space before the number.
1. Factor each polynomial. (See Example 6)
a. 𝑥2
+ 9𝑥 + 18
b. 𝑦2
− 2𝑦 − 24
c. 𝑤2
+ 3𝑤 − 10
2. Factor each polynomial. (See Example 7)
a. 2𝑚2
+ 9𝑚 + 4
b. 6𝑡2
− 𝑡 − 15
YOUR SOLUTIONS:
1. 2. 3.
4. 5.
3. Prepared by: Gauben L. Malicsi
INDICATORS Meets
Standard of
Excellence
Approaching
Standard of
Excellence
Meets
Acceptable
Standard
Does Not Meet
Acceptable
Standard
CRITERIA 4 3 2 1
Performance
Task
Shows
exemplary
performance
Demonstrates
solid
performance
and
understanding
With some
errors and
mastery is
not
thorough.
Has errors,
omission and
misconception.
Assessment
Task
With 5
correct
answers
With 4
correct
answers
With 3
correct
answers
With less than
3 correct
answers
Completeness Has all
aspects of
work that
exceed level
of
expectation
Has some
aspects of
work that
exceed level
of
expectation
Has minimal
aspects of
work that
meet level
of
expectation
No aspect of
work meets
level of
expectations
Neatness The learning
tasks are
done very
neatly.
The learning
tasks are
done neatly.
The learning
tasks are
done quite
neatly.
The learning
tasks are
poorly done
and need
improvement.
Submission of
Requirements
The assigned
learning
tasks are
submitted on
or before
the
deadline.
The assigned
learning
tasks are
submitted a
day after the
deadline.
The assigned
learning
tasks are
submitted
two days
after the
deadline.
The assigned
learning tasks
are submitted
three days
after the
deadline.
TOTAL SCORE
_________________________________ _________________________________
Parent/Guardian’s Signature Over
Printed Name
Student’s Signature Over Printed
Name