🧮 algebra

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

1. مسئله: حل معادله بدون مقدار k داده شده است. 2. ابتدا باید معادله را مشاهده کنیم تا بتوانیم روش حل را انتخاب کنیم.

1. The problem asks to find the domain and range for the "fourth one." Since no explicit function is given, I will explain how to find domain and range generally. 2. **Domain** is

1. The problem is to analyze the function $y = \sqrt{1 - x^2}$. 2. This function represents the upper half of a circle centered at the origin with radius 1, because the equation $x

1. **State the problem:** Solve the simultaneous equations: $$\frac{x}{2} + \frac{y}{3} = 5$$

1. **State the problem:** Solve the exponential equation $$128^x = 32$$ by expressing each side as a power of the same base and then equating exponents. 2. **Identify the bases:**

1. **State the problem:** Expand the logarithmic expression $\log_b(z^6 x)$ using properties of logarithms. 2. **Recall the logarithm properties:**

1. The problem is to evaluate $\log_{10}\left(\frac{1}{10}\right)$ without using a calculator. 2. Recall the logarithm rule: $\log_b\left(\frac{1}{a}\right) = -\log_b(a)$ for any p

1. Problem: Evaluate each expression without using a calculator. 2. Recall the rules of exponents:

1. **State the problem:** Solve the equation $x^2 + 1 = 0$ for $x$. 2. **Recall the formula:** To solve quadratic equations of the form $ax^2 + bx + c = 0$, we use the quadratic fo