🧮 algebra

1. The problem is to simplify the expression $2^4 \times 2^{n+2} - 2 \times 2^{n+2}$ and check if it can be written as $2^4 \times -2 + 2^{n+2} \times 2^{n+2}$. 2. Recall the laws

1. **State the problem:** Given the equation

1. **State the problem:** We have a function defined by the functional equation $$f(x+1) = x - f(x)$$ with the initial condition $$f(1) = 1$$. We need to find $$f(2025)$$. 2. **Und

1. **State the problem:** We have 15 students with exam scores. The mode and median are both 76, the mean score is 80, and the total score of the 7 lowest-scoring students is 515.

1. Let's clarify the concept of exponents. 2. When a number has no visible exponent, it actually means the exponent is 1, not 0.

1. **State the problem:** We are given two equations of lines: $$-7x - 6y = 4$$

1. **State the problem:** Solve the system of linear equations: $$-3y + 4x = 13$$

1. **Problem 1:** Find the quotient and remainder when $6x^4 + x^3 - x^2 + 5x - 6$ is divided by $2x^2 - x + 1$. 2. **Problem 2:** Find the quotient and remainder when $2x^4 + 1$ i

1. **Stating the problem:** Find the sum of the first thirteen terms of the exponential sequence with first term $a_1 = \frac{10 \cdot \binom{11}{6}}{7}$ and common ratio $r = \fra

1. The problem is to understand why we add one to both sides of an equation. 2. When solving equations, the goal is to isolate the variable on one side.

1. **Problem statement:** Determine if each relation defines a function from $\mathbb{R}$ to $\mathbb{R}$. If not, explain why and suggest domain restrictions. 2. **Recall:** A fun

1. **State the problem:** We have a quadratic equation $$x^2 + mx + 58 - m = 0$$ where $m$ is a real number. The roots of this equation are integers. We need to find the largest po

1. Problem 1: Given $p(x) = ax^3 + 5x^2 - 4x + b$, $(x+2)$ is a factor and remainder when divided by $(x+1)$ is 2. 2. Use factor theorem: $p(-2) = 0$ and remainder theorem: $p(-1)

1. **State the problem:** Find values of $a$ and $b$ such that the polynomial $p(x) = 2x^3 + ax^2 - 11x + b$ is divisible by $(2x - 1)$ and leaves remainder 12 when divided by $(x

1. The problem is to solve the equation $2x + 3 = 7$ for $x$. 2. Use the formula for solving linear equations: isolate $x$ by performing inverse operations.

1. **State the problem:** We have a polynomial $p(x) = 2x^3 + ax^2 - 11x + b$. It is given that $p(x)$ is divisible by $(2x - 1)$ and when divided by $(x + 1)$ the remainder is 12.

1. The problem is to generate a graph of a function and provide its answer. 2. Since no specific function was given, I will assume a simple example function: $y = x^2$.

1. **Problem:** Factorise completely: $3x^2 - 2xy - 8y^2$. 2. **Formula and rules:** To factorise a quadratic expression $ax^2 + bx + c$, we look for two numbers that multiply to $

1. **State the problem:** Simplify the expression $$\left(\frac{y+3}{y^2+3y+9}\right) + \left(\frac{y-3}{y^2-3y+9}\right) - \left(\frac{54}{y^4+9y^2+81}\right).$$ 2. **Factor the d

1. **State the problem:** Solve for the two possible values of $a$ in the equation $$\frac{a^2}{2} + 27 = 35.$$\n\n2. **Isolate the term with $a^2$:** Subtract 27 from both sides t

1. **State the problem:** Simplify the expression $$2x \sqrt[3]{16x^{4}} + \sqrt[3]{375x^{8}} - 2x^{3} \sqrt[3]{54x}$$ and express it in the form $$Ax^{2} \sqrt[3]{Bx^{2}} - Cx^{n}