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Questions and Answers
How many rubber bands does she need?
What is the result of dividing (x + x - 2) by (x - 1)?
What remainder is obtained when dividing the polynomial 2x³ - 3x + x + 4 by the binomial x - 2?
If (h + 5) is a factor of h³ + 7h² + 2h - 40, what is the other factor?
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What is the area of a rectangle with a length of (3x - 7) cm and a width of (4x + 5) cm?
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If a man bought a house for Php 800,000 and it appreciates Php 15,000 per year, how much is it worth at the end of 2022?
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How many subscribers does Yumi have after six shares to her friends?
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How many eggs does the farmer collect with the pattern given in the first three baskets?
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What type of sequence is used in Reign's saving plan?
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Which of the following represents an arithmetic sequence?
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Which of the following does NOT illustrate a geometric sequence?
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What are the missing terms in the sequence 6, ___, ___, ___, 22,…?
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What is the 5th term of the sequence described by the formula $a_n = 3n + 8$?
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What is the remainder when a polynomial is divided by $(x + 1)$?
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What expression gives the remainder when $P(x)$ is divided by $(x - 3)$?
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How many marbles did Liam keep on day 12 if he followed the sequence 4, 8, 12, 16…?
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Study Notes
Saving Plans and Sequences
- Reign's saving method involves adding 5 pesos daily, starting with 10 pesos, categorizing it as an Arithmetic Sequence with a common difference of 5.
- Arithmetic sequences are characterized by a constant difference between consecutive terms.
Examples of Sequences
- Classic examples of Arithmetic Sequences include:
- 5, 10, 15, 20... (constant difference of 5)
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Geometric Sequences are identified by a constant ratio, such as:
- 3, 6, 12... where each term is multiplied by 2.
Identifying Sequences
- The sequence 6, ___, ___, ___, 22 can be figured out, with options including:
- 10, 13, 16 as a valid progression with a common difference of 3.
- To find the 5th term of the sequence expressed as ( a_n = 3n + 8 ), substituting ( n = 5 ) yields 23.
Polynomial Division
- Synthetic division is a method used to divide polynomials and to find specific quotients.
- The remainder theorem states that the remainder of a polynomial ( P(x) ) divided by ( (x - c) ) is equal to ( P(c) ).
Specific Calculations
- Liam follows a pattern of buying marbles: 4, 8, 12, 16... leading to him collecting 48 marbles on day 12.
- Identifying the largest common factor for ( 12^2 ) and ( 8^5 ) leads to ( 4^2 ).
Geometric Mean and Quotients
- The geometric mean of -3 and -108 is calculated as -18.
- When dividing ( (x^2 + x - 2) ) by ( (x - 1) ), the result is ( x + 2 ).
Area and Factorization
- The area of a rectangle with length ( (3x - 7) ) cm and width ( (4x + 5) ) cm can be derived from the product of the two expressions.
- Complete factorization of ( 2^d - 32 ) results in ( 4 ).
Real-Life Applications
- A house purchased for Php 800,000 in 2006 appreciates by Php 15,000 yearly, reaching Php 1,040,000 by the end of 2022.
- Mr. Santos saves Php 2,500 monthly on top of an existing Php 15,000, accumulating amounts of Php 17,500, Php 20,000, Php 22,500, and Php 25,000 after successive months.
Egg Collection Problem
- A farmer uses eight baskets with an exponential increase in the number of eggs: three in the first, six in the second, and twelve in the third, doubling in subsequent baskets, culminating in a total collection of 765 eggs.
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Description
This quiz focuses on understanding arithmetic sequences, their characteristics, and examples. Additionally, it covers polynomial division and the remainder theorem, providing a comprehensive assessment of these mathematical concepts.
