🧮 algebra

1. **State the problem:** Given a polynomial $$P(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$$ with non-negative integer coefficients, we know:

1. Factorise fully. (i) Factorise $27y^2 - 3$

1. **State the problem:** We want to find the coefficients $a_0$, $a_1$, and $a_2$ in the power series expansion of $$\left(\frac{1}{1-x}\right)^2 = \sum_{n=0}^\infty a_n x^n$$ whe

1. We are asked to find the coordinates of point A where the line $y = 2x + 1$ intersects the curve $y = \frac{5}{x+1} + 2$. 2. To find the intersection, set the two equations equa

1. **Problem:** Find the coefficients $a_0$, $a_1$, and $a_2$ in the power series expansion $$\frac{1}{(1 - x)^2} = \sum_{n=0}^\infty a_n x^n$$ for $|x| < 1$. **Step 1:** Recall th

1. **State the problem:** We need to find the maximum or minimum points of the quadratic function $$y=x^2+3x-28$$ using symmetry, and then sketch the graph showing all axis interce

1. Find the determinant of matrices: i.

1. **Stating the problem:** The problem involves evaluating a very complex continued fraction with 2025 layers, involving fractions with denominators 113, 242, 355, and so forth, a

1. பிரச்சினையை விளக்குக: 64. 6423 = 36 மற்றும் 2553 = 36 எனப்பட்டுள்ளன. 2. இதில் வேறுபாடு என்ன என்பதை கண்டறிவோம். 6423 மற்றும் 2553 எண்ணிக்கை இரண்டிலும் 36 என்று இருக்கிறது. 3762 இ

1. **Problem 1:** The line $12 = ax + by$ intercepts the coordinate axes at points with $x$-intercept $2$ and $y$-intercept $-3$. Find which of the given points lies on this line.

1. مسئله اول: معادله $$x^4 + (m-3)x^3 + m = 0$$ دارای سه ریشه حقیقی متمایز است و می‌خواهیم مجموع مربعات ریشه‌ها را پیدا کنیم. 2. فرض کنیم ریشه‌ها $$x_1, x_2, x_3, x_4$$ باشند.

1. **State the problem:** We need to find the equation of the directrix of the parabola given by $$y^2 + 4y + 4x + 2 = 0.$$ 2. **Rewrite the equation:** Group the $y$ terms to comp

1. مسئله: داریم معادله $x^2 + 2x - 4 = 0$ که ریشه‌های آن $\alpha$ و $\beta$ هستند. می‌خواهیم معادله‌ای را پیدا کنیم که ریشه‌های آن $2\alpha$ و $2\beta$ باشد. 2. ابتدا رابطه‌های مرب

1. مسئله اول: اگر \( \alpha \) و \( \beta \) ریشه‌های معادله \( 3 = x^3 - x - 2 \) باشند، مقدار \( \beta^4 - 12\alpha^4 \) را پیدا کنید. 2. ابتدا معادله را به شکل استاندارد بنویسیم

1. **State the problem:** Given that $m+n=3$, find the value of the expression $$2m^2 + 4mn + 2n^2 - 6.$$ 2. **Rewrite and simplify the expression:** Notice we can group the terms

1. مسئله: معادله $x^2 + mx + 2 = 0$ دارای دو ریشه حقیقی متمایز و هر دو ریشه منفی است. 2. برای معادله درجه دوم $x^2 + mx + 2 = 0$ ریشه ها را با $x_1$ و $x_2$ فرض کنیم.

1. **State the problem:** We are given positive integers $a$ and $b$ such that $$a^2 + 2ab - 3b^2 - 41 = 0$$ and we need to find the value of $$a^2 + b^2$$. 2. **Rewrite the given

1. The problem states we have a polynomial $$P(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$$ with non-negative integer coefficients $a_i$. 2. Given:

1. Given are several equations involving variables $P$ and $Q$: $P = -4Q + 40000$

1. Problem: Find the negative integer solutions to the inequality $x > -4$. The integers greater than $-4$ are $-3, -2, -1, 0, 1, \ldots$, but the problem asks for negative integer

1. **State the problem:** We are given a function $$f(m,n) = \prod_{k=m}^n \left(1 - \frac{1}{k}\right)$$