🧮 algebra

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}

1. **Stating the problem:** We are given three consecutive terms of an exponential sequence: $$10 \cdot \binom{n}{6} / 7, \binom{b}{7}, \binom{n}{8}$$

1. **Énoncé du problème :** Résoudre l'équation logarithmique $$\ln(x+1) - \ln(x-3) = \ln(6x - 9)$$.

1. **Stating the problem:** We need to solve the equation $$t = \frac{301.5}{v \cos 25^\circ}$$ for $v$ given a value for $t$.

1. **State the problem:** Simplify the expression $$2x \sqrt[3]{16x^4} + \sqrt[3]{375x^8} - 2x^2 \sqrt[3]{54x}$$. 2. **Recall the cube root and exponent rules:**

1. **State the problem:** Solve the equation $7 + 9x - 18 = 1$ for $x$. 2. **Simplify the left side:** Combine like terms $7$ and $-18$.

1. **State the problem:** Simplify the expression $$\frac{2x^2 - 8}{4x}$$. 2. **Recall the formula and rules:** To simplify a rational expression, factor numerator and denominator

1. **State the problem:** Simplify the expression $y\sqrt{20} + \sqrt{80y^2}$ assuming all variables are positive. 2. **Recall the rules:**

1. **Problem:** Sketch the graph of the function $y = -2(x - 5)^2 + 8$. 2. **Formula and rules:** This is a quadratic function in vertex form $y = a(x-h)^2 + k$, where $(h,k)$ is t