🧮 algebra

1. مسئله: باقی‌مانده تقسیم چندجمله‌ای $p(x)$ بر $x+1$ برابر ۲ و بر $x-1$ برابر ۳ است. باید باقی‌مانده تقسیم عبارت $$p(3x-2) + 2x^2 p'(x-2) + x p(3x-2)$$ بر $x-1$ را پیدا کنیم. 2. ق

1. **Problem statement:** Given that the remainders of polynomial $p(x)$ when divided by $x+1$ and $x-1$ are 2 and 3 respectively, find the remainder of the polynomial $$x p(x-2) +

1. The problem involves understanding the division expressions given: \n\nFirst division: divisor = 2, quotient = 2, dividend = a - 1 2 (likely meaning a - 12)\n\nSecond division:

1. مسئله: عبارت $$24 - 9x + 7x^2 + 5x^3$$ بر $$x - 2$$ بخش‌پذیر است. باید این عبارت را تجزیه کنیم. 2. قانون بخش‌پذیری: اگر چندجمله‌ای $$P(x)$$ بر $$x - a$$ بخش‌پذیر باشد، آنگاه $$P

1. **State the problem:** Solve the simultaneous equation \( \log_3(2x + y) = 2 \). 2. **Recall the logarithm definition:** \( \log_b(a) = c \) means \( b^c = a \).

1. The problem is to create two equations. 2. An equation is a mathematical statement that asserts the equality of two expressions.

1. **State the problem:** Solve the quadratic equation in two variables: $5x^2 - 5xy + 3y^2 = 0$. 2. **Formula and approach:** This is a homogeneous quadratic equation. To find sol

1. Simplify $a^{-3}$ using the rule $x^{-n} = \frac{1}{x^n}$: $$a^{-3} = \frac{1}{a^3}$$ 2. Simplify $3x^0$ using the rule $x^0 = 1$ for any $x \neq 0$: $$3x^0 = 3 \times 1 = 3$$

1. The problem is to solve the equation given by the user, but since no specific equation was provided, I will demonstrate solving a simple example equation: $2x + 3 = 7$. 2. The f

1. The problem is to solve the equation given by the user, but since no specific equation was provided, I will demonstrate solving a simple example equation: $2x + 3 = 7$. 2. The f

1. Transform $(9^2)^{1/2}$ to radical form: $$(9^2)^{1/2} = \sqrt{9^2}$$

1. Multiply under one radical: $$\sqrt{8x} \cdot \sqrt{4x} \cdot \sqrt{3x} = \sqrt{8x \cdot 4x \cdot 3x} = \sqrt{96x^3} = \sqrt{16 \cdot 6 \cdot x^3} = 4x\sqrt{6x}$$ 2. Multiply un

1. Problem: Simplify the expression $$\sqrt{8x} \cdot \sqrt{4x} \cdot \sqrt{3x}$$. 2. Formula: Recall that $$\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$$ for any non-negative $a,b$

1. **State the problem:** We need to solve the equation $$2x + 3 = 7$$ for $x$. 2. **Formula and rules:** To solve a linear equation, isolate the variable on one side by performing

1. The problem is to solve the equation $2x + 3 = 7$ for $x$. 2. We use the basic algebraic principle of isolating the variable $x$ by performing inverse operations.

1. The problem is to solve the equation $2x + 3 = 7$ for $x$. 2. We use the basic algebraic principle of isolating the variable $x$ by performing inverse operations.

1. The problem is to solve the equation $2x + 3 = 7$ for $x$. 2. We use the basic algebraic principle of isolating the variable $x$ by performing inverse operations.

1. مسئله: تابع $$f(x) = \frac{ax^2 + bx + c}{(x + d)(x - 3)}$$ به صورت متقابل است. باید مقدار $$a + b - c + d$$ را پیدا کنیم. 2. تابع متقابل یعنی $$f(f(x)) = x$$. این شرط به ما کمک

1. Simplify $8\sqrt{3} - 4\sqrt{3}$. Since both terms have $\sqrt{3}$, subtract the coefficients:

1. Simplify $\sqrt{12}$. We know $12 = 4 \times 3$ and $\sqrt{4} = 2$.

1. The problem is to find the expression for the function $$y = -2 f(1 - x) + 2$$ given the points of the original function $f(x)$: $(-2, 3)$, $(-1, 0)$, $(0, 1)$, and $(1, 0)$.\n\