🧮 algebra

1. Problem: Find the quotient and remainder when dividing the polynomial $$2x^3 + x^2 - x - 4$$ by $$x - 2$$. 2. Use polynomial long division:

1. El problema plantea analizar la función g(x) = x + 2 y la desigualdad \(|x+1|<3\). 2. La desigualdad \(|x+1|<3\) significa que la distancia entre x y -1 es menor que 3, es decir

1. **Problem statement:** Prove by mathematical induction that $$4^2 + 7^2 + 10^2 + \dots + (3n + 1)^2 = \frac{1}{2} n (6n^2 + 15n + 11)$$ for all positive integers $n$. 2. **Base

1. La línea recta que representa la ecuación $y = -3x + 2$ para los valores $x = 2$ y $x = 3$: Calculemos los valores de $y$:

1. Despejar $x$ en la ecuación $\sqrt{x} - 4 = w$. Para despejar $x$, primero sumamos 4 a ambos lados:

1. Despejar la variable $D$ en la ecuación $$A = \frac{Dd}{2}$$: Multiplicamos ambos lados por 2:

1. **State the problems:** 6. Prove the geometric series sum formula: $$a + ar + ar^2 + \cdots + ar^{n-1} = a \frac{1 - r^n}{1 - r}$$

1. The given expression is \( \frac{1-r^k}{1-r} \).\n2. This expression is a common formula in algebra called the sum of a geometric series.\n3. It represents the sum of the series

1. The problem is to simplify the expression $$\frac{1-r^{k-1}}{1-r}$$. 2. Recognize that the denominator is a difference of terms involving $r$, commonly appearing as the denomina

1. Given the identity to prove: $(n \text{ choose } r) + (n \text{ choose } r-1) = (n+1 \text{ choose } r)$. This is a key binomial coefficient identity. 2. Recall the definition o

1. The problem statement "solve for rhs before lhs" is unclear because typically in equations we solve for a variable rather than for "rhs" or "lhs" which stand for right-hand side

1. **Prove by induction that** $a + ar + \cdots + ar^{n-1} = a \frac{1-r^n}{1-r}$ for $r \neq 1$. Step 1: Base case ($n=1$):

1. **Problem Q1:** Express each of the following as an entire surd and write them in ascending order: 3\sqrt{11}, 2\sqrt{23}, 7\sqrt{2}, 10, 4\sqrt{6}. - Convert each to a single s

1. We are given three workers A, B, and C with individual times to complete a tank: A takes 12 days, B takes 15 days, and C takes 20 days. 2. First, find each worker's rate per day

1. **State the problem:** We need to find how long it will take for two pumps working together to empty a tank if the first pump takes 11 hours alone and the second pump takes 20 h

1. Stating the problem: Plumber A can install a bathroom in 12 hours, and Plumber B can do it in 8 hours. We want to find how long it takes if they work together. 2. Calculate indi

1. **State the problem:** We have an arithmetic progression (A.P.) with first term $a_1 = 6$.

1. Stating the problem: The first term ($a_1$) of an arithmetic series is 51, and the eighth term ($a_8$) is 100. We need to find the twentieth term ($a_{20}$). 2. Recall the gener

1. **State the problem:** We need to find the sum of the first 30 terms of the arithmetic sequence: 4, 7, 10, 13, ... 2. **Identify the sequence details:**

1. **Problem statement:** Find the sum of the first 30 terms of the arithmetic sequence: 4, 7, 10, 13, ... 2. **Identify the sequence characteristics:**

1. **State the problem:** Cindy was asked to subtract 3 from a certain number $x$ and then divide the result by 9. 2. The correct expression for the task is $$\frac{x - 3}{9}.$$