🧮 algebra

1. **State the problem:** We are given two equations:

1. Pernyataan masalah: Buktikan bahwa untuk setiap $n \geq 1$, berlaku rumus jumlah kuadrat berikut: $$1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}$$

1. **State the problem:** Factor the quadratic expression $X^2 + 7X + 12$. 2. **Recall the factoring formula:** For a quadratic $X^2 + bX + c$, we look for two numbers that multipl

1. **State the problem:** Solve for $x$ in the equation $$\frac{-3(x+7)}{-7x+9} = \frac{1}{6}.$$\n\n2. **Understand the equation:** We have a rational equation where a fraction equ

1. Diketahui matriks $A = \begin{pmatrix} 3 & -1 \\ 11 & -4 \end{pmatrix}$. Kita diminta mencari invers matriks $A$, yaitu $A^{-1}$. 2. Rumus invers matriks 2x2 adalah $$A^{-1} = \

1. **State the problem:** Find the remainder when the expression $\left(6^3 + 4^2 \times 2^1\right)$ is divided by 8. 2. **Write the expression:**

1. **State the problem:** Identify vertical asymptotes, horizontal asymptotes, and holes of the function $$f(x) = \frac{(2x - 6)(x + 1)}{(x + 1)(x - 1)}.$$\n\n2. **Simplify the fun

1. **State the problem:** We need to find the vertical asymptotes, horizontal asymptotes, and holes of the function $$f(x) = \frac{(x+1)(x+2)}{(x+2)(x-3)}.$$\n\n2. **Identify verti

1. **بيان المسألة:** لدينا دالة كثيرة الحدود من الدرجة الثالثة بالشكل $f(x) = ax^3 + 6x^2 + bx + 4$، ونريد إيجاد القيم $a$ و $b$ بحيث يمر المنحنى $(C)$ بالنقطة $A(1,0)$ ويقبل مماسً

1. **State the problem:** Divide the polynomial $4x^4 - 33x^2 - 9x + 2$ by $p(x) = x - 3$ using polynomial long division. 2. **Recall the division formula:** For polynomials, divid

1. نبدأ بتحديد مجموعة تعريف كل دالة \(f\) و \(g\) في الحالة الأولى: - \(f(x) = 2x - 1\) هي دالة خطية، ومجموعة تعريفها هي جميع الأعداد الحقيقية \(\mathbb{R}\).

1. **State the problem:** Find the domain, vertical asymptote(s), and horizontal asymptote(s) of the function $$f(x) = \frac{4 - 2x}{7x - 1}$$.

1. نبدأ بذكر المشكلة: هل الدالتان $f$ و $g$ متساويتان في كل حالة من الحالات المعطاة؟ 2. القاعدة الأساسية لمساواة دالتين هي أن تكون لهما نفس المجال ونفس القيم لكل $x$ في هذا المجال.

1. **State the problem:** Simplify or analyze the expression $4x^4 - 33x^2 - 9x + 2$. 2. **Identify the expression:** This is a polynomial of degree 4 with terms $4x^4$, $-33x^2$,

1. **State the problem:** Find the domain, vertical asymptotes, and horizontal asymptotes of the function $$f(x) = \frac{x}{x^2 - 1}$$. 2. **Domain:** The domain of a function incl

1. **State the problem:** We are asked to divide the polynomial function $f(x) = 4x^4 - 33x^2 - 9x + 20$ by $p(x) = x - 3$. 2. **Recall the division formula:** When dividing a poly

1. **State the problem:** We are given the rational function $f(x) = \frac{5}{x - 4}$ and need to analyze its domain, vertical and horizontal asymptotes. 2. **Domain:** The functio

1. **State the problem:** We need to find the quotient and remainder when dividing the polynomial $$f(x) = 4x^4 - 33x^2 - 9x + 20$$ by $$x - 3$$. 2. **Formula and rule:** Polynomia

1. **State the problem:** We need to divide the polynomial $$f(x) = -33x^3 - 9x + 2$$ by the polynomial $$p(x) = x - 3$$ and find the quotient and remainder. 2. **Formula and rules

1. Enunțul problemei: Trebuie să rezolvăm ecuația $$x^5 + 80x - 10 = 0$$ pe mulțimea numerelor reale $$\mathbb{R}$$ folosind principiul contracției, cu o aproximație de $$10^{-1}$$

1. **State the problem:** Find the constant of proportionality $k$ given that $z$ is directly proportional to the product of $x$ and the cube root of $y$, and when $x=2$, $y=8$, $z