🧮 algebra

1. **State the problem:** Solve the cubic equation $$6x^3 + 25x^2 - 24x + 5 = 0$$ given that one rational root is 5. 2. **Use the Factor Theorem:** Since 5 is a root, $(x - 5)$ is

1. **Problem:** Show that if the linear equations $x_1 + kx_2 = c$ and $x_1 + lx_2 = d$ have the same solution set, then the two equations are identical (i.e., $k = l$ and $c = d$)

1. **Problem Statement:** Given two functions $f(x) = 2x$ and $g(x) = x + 9$, find: a. $(f \circ g)(x)$

1. **State the problem:** Solve the system of equations: $$\begin{cases} 2x + 3y - z = 5 \\ x - y + 4z = 2 \\ 3x + 2y + z = 7 \end{cases}$$

1. The problem is to factor the expression $x^2 - 16$. 2. Recognize that this is a difference of squares, which follows the formula $a^2 - b^2 = (a - b)(a + b)$.

1. The question asks about the hardest equation to factor. 2. Factoring difficulty depends on the type and degree of the polynomial.

1. The problem: Understand the basic rules for factoring algebraic expressions. 2. Factoring is the process of breaking down an expression into simpler expressions (factors) that,

1. The problem asks which graph represents the solution set of the inequality $y > 4x - 3$. 2. The boundary line is given by the equation $y = 4x - 3$.

1. **State the problem:** We need to find the graph that best represents the solution set for the system of inequalities: $$y \leq -\frac{2}{3}x - 2$$

1. The problem is to simplify the expression $-9-9$. 2. This is a subtraction problem involving two integers.

1. The problem is to evaluate the expression $-1 + [-4]$. 2. The brackets around $-4$ indicate it is a single number, so the expression simplifies to $-1 + (-4)$.

1. The problem is to determine if a certain expression is equivalent to $\frac{\log 5}{\log 4}$. 2. Recall the change of base formula for logarithms:

1. **State the problem:** Simplify the expression $$\frac{20}{\log 4^2 - \log 4}$$. 2. **Recall logarithm properties:**

1. **State the problem:** Simplify the expression $$\frac{3^{8} \cdot 2^{7} \cdot 4^{2} \cdot 25^{2}}{3^{6} \cdot 2^{5} \cdot 2^{4} \cdot 5^{4}} - \frac{243^{\frac{1}{2}}}{9^{\frac

1. **Problem statement:** We are given two linear equations: $$x_1 + kx_2 = c$$

1. **State the problem:** Solve the equation $3(5x-29)-9=-6$ for $x$. 2. **Apply the distributive property:** Multiply 3 by each term inside the parentheses:

1. **State the problem:** Solve the equation $$\frac{1}{2}r - \frac{3}{4} = \frac{1}{2}$$ for $r$. 2. **Isolate the variable term:** Add $\frac{3}{4}$ to both sides to get rid of t

1. **Problem statement:** We have a battery level function $$B(t) = 100\left(1 - \frac{t}{10} e^{-0.2t}\right)$$ for $$t \in [0,10]$$ hours. We need to find: a. The maximum and min

1. **State the problem:** We have a piecewise function: $$f(x) = \begin{cases} 2x + 1 & \text{if } x \leq 2 \\ -4 & \text{if } x > 2 \end{cases}$$

1. The problem is to understand the structure of the ring $Q[x,y]$, which is the ring of polynomials in two variables $x$ and $y$ with coefficients in the rational numbers $Q$. 2.

1. **State the problem:** We are given the function $f(x) = (x - 1)^2 + 2$ and need to find its domain and range. 2. **Domain:** The domain of a function is the set of all possible