🧮 algebra

1. Stating the problem: We need to divide the polynomial $$6x^4 - 4x^3 + 8x^2 - 2x + 1$$ by the binomial $$3x + 1$$. 2. Set up the long division: We are dividing $$6x^4 - 4x^3 + 8x

1. The problem is to convert ?2,900 to US dollars given the exchange rate 1 USD = ?58.00. 2. To find how many US dollars correspond to ?2,900, we divide the amount in the local cur

1. We are given the equation $$\sqrt[4]{8} \cdot 625 \cdot 81 \cdot \Box = \Box \cdot 5 \cdot 2$$ where \Box represents unknown values that are the same on both sides. 2. To solve

1. We start with the expression: $$4y(3x-1)+(3x-1)^2$$ 2. Notice that both terms contain the factor $(3x-1)$.

1. The problem asks whether the expression $(q^2-2)x^2(-9x^6y^7+11)$ can be written as $x^2[(q^2-2)(-9x^6y^7+11)]$. 2. Start by distributing $x^2$ inside the parentheses in the sec

1. Stating the problem: We need to verify if the expression $(q^2-2)x^2(-9x^6y^7+11)$ can be rewritten as $x^2[(q^2-2)x^2(-9x^6y^7+11)]$. 2. Start with the original expression:

1. **State the problem:** Factorise the expression $$-9x^8 y^7 (q^2 - 2) + 11 x^2 (q^2 - 2)$$. 2. **Identify common factors:** Notice that the binomial $$q^2 - 2$$ appears in both

1. **State the problem:** Solve the inequality $$x^3 - 2x^2 + 5x + 20 \geq 2x^2 + 14x - 16$$ and analyze the sign of $$(x - 4)(x - 3)(x + 3) \geq 0$$.

1. The problem gives a 4x4 matrix: $$\begin{pmatrix} 6 & -5 & 1 & -3 \\ 2 & -4 & 8 & 3 \\ 4 & -7 & -6 & 5 \\ -2 & 9 & 7 & -1 \end{pmatrix}$$

1. Solve each inequality and express the solution in interval notation. (a) Solve $$-4[x + 2(3 - x)] \leq 5(2 - 4x) + 6$$

1. The problem is to draw the feasible regions for the systems of linear inequalities given in parts a, b, and c. These regions are bounded by lines and axes. 2. Part (a): 2x + y \

1. **State the problem:** We are asked to graph the solution regions of three systems of linear inequalities. 2. **Part a:** The inequalities are:

1. Let's start by understanding the logarithm (log). The logarithm answers the question: "To what power must we raise a certain base to get a number?" 2. The logarithm with base $b

1. The user's message is just "log," which is a general term related to logarithms. 2. To explain logarithms briefly: The logarithm base $b$ of a number $x$ answers the question: "

1. El problema requiere simplificar la expresión $$(x - 2y^3)^2 (x^3 y^{-1}).$$ 2. Primero, expandimos el cuadrado del binomio $$(x - 2y^3)^2$$ usando la fórmula de cuadrado de dif

1. Planteamos el sistema de ecuaciones: $$5x - y = 3$$

1. The problem is to find the zeros (roots), extrema, and intercepts of the quadratic function $$y = x^2 - 5x + 6$$. 2. First, find the zeros by setting $$y = 0$$:

1. We are asked to divide the polynomial $15x^4 - 25x^3 - 36x^2$ by $5x^2$. 2. We will divide each term of the numerator by $5x^2$ separately:

1. Stating the problem: Simplify the expression $$\frac{15x^4 - 25x^3 - 36x^2}{5x^2}$$. 2. Factor the numerator where possible: The numerator is $$15x^4 - 25x^3 - 36x^2$$.

1. The problem is to find a formula for the sum of the squares of the first $n$ natural numbers, i.e., calculate $$\sum_{i=1}^n i^2.$$\n\n2. We observe the series: $1^2 + 2^2 + 3^2

1. Stating the problem: Solve the inequality $$8 < 2x < 12$$ for the variable $x$. 2. Break the compound inequality into two parts: