🧮 algebra

1. Let's start by defining a **sequence**. A sequence is an ordered list of numbers, where each number is called a term. For example, the sequence $1, 3, 5, 7, \dots$ lists odd num

1. Let's start by defining what a sequence and a series are. 2. A sequence is an ordered list of numbers, for example, $a_1, a_2, a_3, \dots, a_n$.

1. **State the problem:** Factor the quadratic expression $x^2 + 8x + 16$. 2. **Identify coefficients:** The quadratic is in the form $ax^2 + bx + c$ where $a=1$, $b=8$, and $c=16$

1. The problem is to factor the quadratic expression $x^2 + 8x + 6$. 2. We look for two numbers that multiply to $6$ (the constant term) and add to $8$ (the coefficient of $x$).

1. **State the problem:** Factor the cubic polynomial $$x^3 + 8x + 16$$. 2. **Look for rational roots using the Rational Root Theorem:** Possible roots are factors of 16 (constant

1. The problem is to factor the expression $x^2 + 8 + 16$. 2. First, notice that the expression is $x^2 + 8 + 16$, which seems to be missing an $x$ term to be a perfect square trin

1. Let's start by defining a **sequence**. A sequence is an ordered list of numbers following a specific pattern. For example, $1, 2, 3, 4, 5, \dots$ is a sequence where each numbe

1. **State the problem:** We are given the equation $$\frac{x^2}{2} + \frac{1}{x^2} - \frac{2}{x}$$ and told it can be rewritten as $$x^4 + ax^3 + bx^2 + cx + 2 = 0.$$ We need to f

1. The problem is to find the equation of the line passing through the points $(1,2)$ and $(3,8)$. 2. First, calculate the slope $m$ using the formula:

1. The problem is incomplete as only "The equation" is provided without further details. 2. Please provide the full equation or problem statement so I can assist you step-by-step.

1. For the linear equation $y = 3 - 4x$: - The equation is already in slope-intercept form $y = mx + b$ where $m$ is the gradient and $b$ is the y-intercept.

1. **State the problem:** Solve the inequality $$-2(4 + 3x) > 7$$ and express the solution in interval notation. 2. **Distribute the -2:**

1. **State the problem:** We are given the equation $$V = \frac{1}{U} + \frac{1}{W}$$ and asked to solve for $U$. 2. **Isolate the term with $U$:** Start by subtracting $$\frac{1}{

1. The problem is to solve for $U$ in an equation where $U$ is the unknown variable. 2. Since the equation is not provided, let's consider a general approach: isolate $U$ on one si

1. The problem is to understand and analyze the equation $$\frac{1}{V} = \frac{1}{U} + \frac{1}{W}$$ which relates three variables $V$, $U$, and $W$. 2. This equation can be interp

1. **State the problem:** The tide starts 20 meters from the shore and recedes at a rate of 3 meters per hour for 6 hours. 2. **Identify what is asked:** Find the distance of the t

1. **State the problem:** We are given a point $a(-4,1)$ and a slope $m=\frac{2}{3}$. We want to find the $y$-coordinate of point $b$ when $x=8$ on the line passing through $a$ wit

1. Let's analyze the expression $(-c^3)^2$. When raising a power to another power, we multiply the exponents:

1. **State the problem:** Solve the equation $$3 |3x - 7| - 1 = 6x + 20$$ for $x$. 2. **Isolate the absolute value:** Add 1 to both sides:

1. Solve the equation $\frac{1}{3}(x+1) = \frac{1}{2}(x-2)$. Multiply both sides by 6 (the least common multiple of 3 and 2) to clear denominators:

1. **Problem statement:** Simplify each expression by applying the power of a product rule and the power of a power rule. 2. **Part a:** Simplify $ (3 m^4)^2 $.