🧮 algebra

1. The function given is $$g(x) = -x^4 + 2x^3 + 5x^2 - 1$$. 2. To determine the end behavior of the polynomial, we focus on the leading term of highest degree because it dominates

1. The problem is to use the Gauss Elimination method to solve the system of linear equations: $$a x + b y + c z = j$$

1. The user requests to write a solution as it would be done on paper, implying detailed, step-by-step work. Since no specific problem is given, here is a demonstration of solving

1. **State the problem:** Given the system of equations:

1. Find the discount rate for a watch originally priced at 5990 and now on sale for 5091.50. Calculate the discount amount:

1. The problem is unspecified in the prompt, so let's outline general steps to tackle algebra problems. 2. Identify what is given and what you need to find.

1. Stating the problem: We are given the polynomial function $$R(z) = z^4 - z^2 + 44z + 26$$. We want to analyze this function, find its critical points, and describe its behavior.

1. State the problem: Solve the quadratic equation $x^2 - 12x = -27$. 2. Move all terms to one side to set the equation to 0:

1. Stating the problem: We are given an original price of $550000 and a discounted price of $350000. We want to find the discount amount and the discount percentage. 2. Calculate t

1. The problem asks to factor the quadratic expression $y = t^2 + 2t - 48$. 2. To factor, we look for two numbers that multiply to the constant term $-48$ and add to the coefficien

1. The problem is to find the percentage discount from 550,000. 2. To find a percentage discount, we need to know the discount amount or the new price after the discount.

1. Problem: Find the original price if the discount is 8% and the discount amount is 55.60. Step 1: Let the original price be $x$.

1. **State the problem:** We are given the graph of function $f(x)$, which forms a "V" shape centered at the origin $(0,0)$. We want to compare $f(x)$ to the function $$g(x) = -2|x

1. Stating the problem: Solve the equation $$e^{x+y} - 3xy - 2 = y$$ for the relationship between $$x$$ and $$y$$. 2. Rearrange the equation to isolate terms:

1. The problem involves the term $-3xy$ as part of an equation or expression. 2. If you provide the full equation, I can help solve or simplify it involving the term $-3xy$.

1. The original function is given as $f(x) = -x^2$. 2. We apply a vertical stretch to $f(x)$ by a factor of 8. A vertical stretch multiplies the $y$-values by the stretch factor.

1. The original function given is $f(x) = |x|$. 2. The transformed function is $g(x) = |x - 5|$.

1. The problem is to find the value of $2000 \times 80000$ in standard form. 2. First, express the numbers in scientific notation:

1. **Problem:** Solve the equation $\frac{5-x}{2x} - \frac{3-2x}{x} = 1$ using the quadratic formula. 2. **Rewrite the equation:** Find a common denominator $2x$:

1. **State the problem:** Two acid solutions with concentrations 25% and 55% are mixed in ratio $m:n$ to form mixture $M$.

1. **Problem statement:** Find the number of ordered integer pairs $(x,y)$ satisfying the equation $$\frac{5}{x} + \frac{1}{y} = \frac{1}{18}.$$\n\n2. **Rewrite the equation:** Mul