Edwin Ogie Library is a dynamic platform for education, focused on fostering mindful communication and building positive relationships by eliminating linguistic errors. Our mission is to enhance connections through thoughtful language, emotional regulation, and self-awareness, providing educational resources that inspire personal growth. We aim to promote well-being, peace, and meaningful connections, offering a space for individuals committed to refining their communication skills.
JSS2 Matter Note with CBT Quiz | Edwin Ogie Library Comprehensive Note on Matter for JSS 2 Well-structured lesson notes, examples, class activity, evaluation, and a 30-question timed CBT quiz to support learning. Topic: Matter Meaning of Matter Matter is anything that has mass and occupies space. Everything around us that can be seen, touched, or measured is matter. Examples Water, air, wood, stone, iron, book, human beings, plants. Characteristics Has mass, occupies space, can be measured, exists in different states. Matter is present in all objects around us, whether living or non-living. States of Matter States of Matter Matter exists mainly in three states: solid, liquid and gas. 1. Solid Definite shape and definite volume. Particles are packed clos...
Get link
Facebook
X
Pinterest
Email
Other Apps
Polynomial Further Mathematics Web Book | Edwin Ogie Library
Polynomial Further Mathematics Web Book | Edwin Ogie Library
Polynomial Further Mathematics Web Book
A complete guide to polynomial operations, factor theorem, remainder theorem, and polynomial division, with worked examples, practice exercises, and a 30-question CBT quiz timed for 15 minutes.
A polynomial is an algebraic expression made of variables, coefficients and non-negative integer powers. Examples include 2x + 3, x² - 4x + 7, and 5x³ - 2x + 1.
Polynomial topics include addition, subtraction, multiplication, division, factor theorem and remainder theorem.
Degree
The highest power of the variable.
Terms
Parts separated by + or −.
Coefficient
The numerical factor of a term.
Remember: a polynomial does not contain negative powers of the variable or the variable in the denominator.
Topic 1
Addition of Polynomials
To add polynomials, combine like terms only. Like terms have the same variable and the same power.
Worked examples
Example 1
(2x + 3) + (4x + 5)
Group like terms: 2x + 4x = 6x, 3 + 5 = 8.
Answer: 6x + 8
Example 2
(3x² + 2x - 1) + (x² - 5x + 4)
Combine x² terms, x terms and constants.
Answer: 4x² - 3x + 3
Example 3
(5a² - a + 7) + (2a² + 3a - 9)
Answer: 7a² + 2a - 2
Topic 2
Subtraction of Polynomials
To subtract polynomials, change the signs of the second polynomial and then combine like terms.
Worked examples
Example 1
(5x + 6) - (2x + 1)
Change signs: 5x + 6 - 2x - 1
Answer: 3x + 5
Example 2
(4x² + 3x - 2) - (x² - x + 5)
Change signs and collect terms.
Answer: 3x² + 4x - 7
Example 3
(6a - 2a² + 7) - (3a + a² - 1)
Answer: -3a² + 3a + 8
Topic 3
Multiplication of Polynomials
Multiply each term in one polynomial by each term in the other. Then simplify by combining like terms.
Worked examples
Example 1
(x + 2)(x + 3)
x·x = x², x·3 = 3x, 2·x = 2x, 2·3 = 6
Answer: x² + 5x + 6
Example 2
(2x - 1)(x + 4)
Answer: 2x² + 7x - 4
Example 3
(3a + 2)(a - 5)
Answer: 3a² - 13a - 10
Example 4
(x - 3)²
Answer: x² - 6x + 9
Topic 4
Division of Polynomials
Polynomials can be divided by another polynomial using long division or synthetic division when applicable. The dividend is divided by the divisor to obtain a quotient and sometimes a remainder.
Worked examples
Example 1
(x² + 5x + 6) ÷ (x + 2)
Answer: x + 3
Example 2
(x² - 9) ÷ (x - 3)
Answer: x + 3
Example 3
(2x² + 7x + 3) ÷ (x + 3)
Answer: 2x + 1
If the divisor is of the form x - a, synthetic division or the factor theorem can be especially helpful.
Topic 5
Remainder Theorem
The remainder theorem states that when a polynomial f(x) is divided by x - a, the remainder is f(a).
Remainder Theorem: If f(x) is divided by x - a, remainder = f(a).
Worked examples
Example 1
Find the remainder when f(x)=x²+3x+1 is divided by x-2.
Compute f(2) = 4 + 6 + 1 = 11.
Answer: 11
Example 2
Find the remainder when f(x)=2x³-x+4 is divided by x+1.
Compute f(-1) = -2 + 1 + 4 = 3.
Answer: 3
Example 3
Find the remainder when f(x)=x³-4x²+2x-5 is divided by x-3.
Compute f(3) = 27 - 36 + 6 - 5 = -8.
Answer: -8
Topic 6
Factor Theorem
The factor theorem states that x - a is a factor of a polynomial f(x) if and only if f(a)=0.
Factor Theorem: x - a is a factor of f(x) when f(a) = 0.
Worked examples
Example 1
Show that x - 2 is a factor of f(x)=x²-5x+6.
Compute f(2) = 4 - 10 + 6 = 0.
Answer: Yes, it is a factor.
Example 2
Test whether x + 1 is a factor of x³ + x² - x - 1.
Compute f(-1) = -1 + 1 + 1 - 1 = 0.
Answer: Yes, it is a factor.
Example 3
Find k if x - 3 is a factor of x² + kx - 12.
Set f(3)=0: 9 + 3k - 12 = 0.
So, 3k - 3 = 0, hence k = 1.
Example 4
Find the factor of x² - 7x + 12.
It factorizes to (x - 3)(x - 4).
Answer: x - 3 and x - 4 are factors.
Practice
20 Exercises
Try the exercises first, then reveal the answers to check your work.
1. Define a polynomial.
Answer: An algebraic expression with non-negative integer powers of the variable.
2. Find the degree of 5x³ - 2x + 1.
Answer: 3
3. Add (2x + 3) and (4x + 1).
Answer: 6x + 4
4. Subtract (3x + 5) from (7x + 2).
Answer: 4x - 3
5. Multiply (x + 1)(x + 5).
Answer: x² + 6x + 5
6. Expand (x - 2)².
Answer: x² - 4x + 4
7. Divide (x² + 7x + 12) by (x + 3).
Answer: x + 4
8. State the remainder theorem.
Answer: The remainder when f(x) is divided by x - a is f(a).
9. Find the remainder when x² + 4x + 1 is divided by x - 1.
Answer: 6
10. Test if x - 2 is a factor of x² - 5x + 6.
Answer: Yes, because f(2)=0.
11. Find the remainder when 2x³ - x + 4 is divided by x + 1.
Answer: 3
12. Factorize x² - 9.
Answer: (x - 3)(x + 3)
13. Factorize x² + 5x + 6.
Answer: (x + 2)(x + 3)
14. Find k if x - 1 is a factor of x² + kx - 2.
Answer: k = 1
15. Find the degree of 3x⁴ + x² - 6.
Answer: 4
16. Multiply (2x - 3)(x + 4).
Answer: 2x² + 5x - 12
17. Subtract (x² - 2x + 1) from (3x² + x - 4).
Answer: 2x² + 3x - 5
18. What is the coefficient of x² in 7x² - 4x + 9?
Answer: 7
19. If f(3)=0, what does that tell you?
Answer: x - 3 is a factor of f(x).
20. Find the quotient of (x² - 4x + 4) ÷ (x - 2).
Answer: x - 2
CBT Quiz
30 questions | 15 minutes | One question at a time
FORGIVENESS THE SECRET TO A SUCCESSFUL RELATIONSHIP Forgiveness is a critical aspect of human relationships, particularly in marriage. Treating others with mercy and leniency requires you to overlook misunderstandings and the wrongs they have done to you. The capacity to forgive is a fundamental virtue that is the linchpin of safeguarding relationships and leading a flourishing life. Lack of forgiveness causes lots of damage to marriages, and it is often the root cause of divorce in our society. It is, therefore, vital to understand that forgiveness is the key to a lasting, successful, and fulfilling marriage. Before embarking on the journey of marriage, individuals should take responsibility and learn to know themselves deeply. It is essential to realize that marriage is not a walk in the park. It requires perseverance, tenacity, and a loving spirit from the partners involved. When hurt arises, forgiveness is an indispensable tool that partners must employ to calm their difference...
Mastering the Art of Present Steps for Future Triumphs" In the pursuit of victory, every step forward is pivotal. It's essential to navigate your present position with care, for it is the springboard to your future achievements. How you handle your current endeavors directly impacts the path to your next conquests. Attempting to bypass or disregard your present circumstances can undermine your success in the steps that follow. In the realm of business, strategic maneuvering is essential to attaining your desired goals. Step into Consciousness: Harnessing Experience for Future Benefits Experience serves as our greatest teacher, guiding us towards tomorrow's rewards. Often, we fall into the trap of assuming the world revolves around us, treating our current situations with indifference and overlooking their implications for future outcomes. It's crucial to be mindful of how we approach our present steps, recognizing their significance in shaping tomorrow's victories....
EO Navigating Life’s Complexities Through Self-Consciousness A practical, story-driven guide to building mental resilience, clarity and better relationships in daily life. Expand all Collapse all Print Download .doc Introduction What is Self-Consciousness? Why it Matters The Nigerian Context Stories in Action Root Causes Upbringing & Environment Strategies to Grow Applications Exercises & Plans Scripts & Conversation Frames Resources Conclusion Introduction In a world of constant noise, stress, and social pressure, developing self-consciousness — the steady ability to notice yourself and choose — is essential. This post explains what it is, why it matters (especially in Nigeria), and how to build it with concrete stories...
Comments
Post a Comment
We’d love to hear from you! Share your thoughts or questions below. Please keep comments positive and meaningful, Comments are welcome — we moderate for spam and civility; please be respectful.