🧮 algebra

1. Newton's identities relate the power sums of the roots of a polynomial to its coefficients. 2. For a polynomial $x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0 = 0$ with roots $r_1

1. **State the problem:** We are given a quartic equation $$x^4 - 2x^3 + x^2 - 4 = 0$$ with roots $a, \beta, \gamma, \delta$. We want to show that $$S_4 = 9S_3$$ where $$S_k = a^k

1. The problem is to find the solution region for the system of inequalities: $$5x + 2y \geq 10$$

1. **Problem Statement:** Given two points on a line with coordinates $(x_1, y_1)$ and $(x_2, y_2)$, we need to find: a) The vertical change between the points.

1. The problem is to simplify the expression $$\sqrt{\frac{8b^8}{49}}$$ assuming $$b > 0$$. 2. We start by using the property of square roots that $$\sqrt{\frac{A}{B}} = \frac{\sqr

1. Problem: Given two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line: a) The vertical change is the difference in the $y$-coordinates: $$\Delta y = y_2 - y_1$$

1. The problem is to simplify the expression $$880 - 800 \left(1 - \frac{\sqrt{s}}{\sqrt{v}}\right)$$. 2. Start by distributing the 800 inside the parentheses:

1. Problem: Solve the equation $1/x = 2$ for $x$. 2. Domain: We require $x\neq 0$ because division by zero is undefined.

1. The problem is to find the zeroes of a function, which means finding the values of $x$ where the function equals zero. 2. Since the function is not specified, let's assume a gen

1. **State the problem:** Factor the polynomial $$5x^5 - 20x^4 + 5x^3 + 50x^2 - 20x - 40$$. 2. **Factor out the greatest common factor (GCF):** Each term is divisible by 5, so fact

1. **بيان المسألة:** لدينا الدالة $$y = f(x) = x^3 - 3x$$ ونريد إيجاد مجال الدالة، مدى الدالة، دراسة الإترداد (زوجية، فردية، أو غير ذلك)، ونقطة العاتق. 2. **مجال الدالة:** الدالة ك

1. Bài toán yêu cầu vẽ đồ thị hàm số $y=\log(x+4)-3$.\n\n2. Hàm số này là hàm logarit cơ bản $\log(x)$ nhưng có sự dịch chuyển: \n- Dịch chuyển ngang sang trái 4 đơn vị vì bên tron

1. **Solve** $\frac{x^2}{3} = 9$. Multiply both sides by 3:

1. The problem asks to solve question number (5) related to simple interest calculations involving principal (P), rate (r%), and time. 2. Simple interest formula is given by $$SI =

1. **State the problem:** We need to find the linear equation that relates $x$ and $y$ based on the given table: $$\begin{array}{c|c} x & y \\ \hline 5 & 40 \\ 6 & 41 \\ 7 & 42 \\

1. The problem provides a set of points $(x, y)$: $(0,0)$, $(1,-2)$, $(2,-4)$, and $(3,-6)$. 2. We want to find the equation of the line that fits these points.

1. **State the problem:** We need to find the linear equation $y = mx + b$ that fits the given table of values: | x | y |

1. **State the problem:** We are given a table of values for $x$ and $y$ and need to find the linear equation $y = mx + b$ that fits the data. 2. **Identify points:** From the tabl

1. We are given a table of values for $x$ and $y$: $$\begin{array}{c|c} x & y \\ \hline 4 & 0 \\ 5 & 1 \\ 6 & 2 \\ 7 & 3 \end{array}$$

1. The problem is to understand the transformation represented by the equation $2y = f(x) - 4$. 2. Start by isolating $y$ to express it in terms of $f(x)$:

1. We are asked to simplify the expression $$\sqrt{\frac{49u^4}{2}}$$ assuming $$u > 0$$. 2. Recall that the square root of a fraction is the fraction of the square roots: $$\sqrt{