Arithmetic Sequences and Polynomial Division

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?

What is the area of a rectangle with a length of (3x - 7) cm and a width of (4x + 5) cm?

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?

How many subscribers does Yumi have after six shares to her friends?

How many eggs does the farmer collect with the pattern given in the first three baskets?

What type of sequence is used in Reign's saving plan?

Which of the following represents an arithmetic sequence?

Which of the following does NOT illustrate a geometric sequence?

What are the missing terms in the sequence 6, ___, ___, ___, 22,…?

What is the 5th term of the sequence described by the formula $a_n = 3n + 8$?

What is the remainder when a polynomial is divided by $(x + 1)$?

What expression gives the remainder when $P(x)$ is divided by $(x - 3)$?

How many marbles did Liam keep on day 12 if he followed the sequence 4, 8, 12, 16…?

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)
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.

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.