🧮 algebra

1. **State the problem:** Factor and graph the polynomial $f(x) = (x^3 + x^2)(3x^2 - 2x - 1)$.\n\n2. **Understand the expression:** The polynomial is given as a product of two poly

1. **State the problem:** We need to expand and simplify the expression $$(x^3+x^2)(3x^2-2x-1).$$ 2. **Formula and rules:** To expand, use the distributive property (also called FO

1. Let's start by understanding what a polynomial is. A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, multiplication,

1. **State the problem:** Solve the equation $25(2^{\log x}) = x$ for $x$. 2. **Recall the properties:** Here, $\log$ denotes the logarithm base 10. We use the property $a^{\log b}

1. The problem is to evaluate the expression $27^{-\frac{2}{3}}$ and simplify it. 2. Recall the rule for exponents: $a^{-b} = \frac{1}{a^b}$ and the fractional exponent rule: $a^{\

1. The problem is to simplify the expression $k \times -5$. 2. The multiplication of a variable by a negative number follows the rule: multiplying by a negative number changes the

1. **State the problem:** We need to calculate $1.02 \times 10^{16} - 7 \times 10^{14}$ and express the answer in standard form. 2. **Recall the standard form:** A number in standa

1. **State the problem:** Find the domain and range of the function $f(x) = x^2$. 2. **Recall definitions:**

1. **Problem Statement:** Find the domain and range of the function $f(x) = x^2$. 2. **Understanding the function:** The function $f(x) = x^2$ is a quadratic function where the out

1. Problem Q1: Find the domain and range of $f(x)=x^2$. 2. Formula and rules: For polynomial functions there are no denominators or even roots to restrict domain, so domain is all

1. **State the problem:** We are given the function $f(x) = kx^2$ and the equation $f(-5k) = 675$. We need to find the value of $k$. 2. **Write the formula and substitute:** The fu

1. **Problem:** Find the domain and range of $f(x) = x^2$. 2. **Domain:** Since $f(x) = x^2$ has no denominators or square roots, all real numbers are allowed. Thus, domain is $\ma

1. **State the problem:** We are given the function $f(x) = kx^2$ and the equation $f(-5k) = 675$. We need to find the value of $k$. 2. **Write the formula and substitute:** The fu

1. **State the problem:** Simplify the expression $$(3 y^{2} - 2 y - 7) + (4 y^{2} - 6 y + 1)$$ by combining like terms. 2. **Formula and rules:** When adding polynomials, combine

1. **State the problem:** Calculate $9.7 \times 10^5 + 9.2 \times 10^6$ and express the answer in standard form. 2. **Recall the standard form:** A number in standard form is writt

1. **State the problem:** We are given two functions: $$f(x) = \frac{x}{x - 9}$$

1. **State the problem:** Expand and fully simplify the expression $x(4x - 2) + x(3x + 7)$. 2. **Recall the distributive property:** For any terms $a$, $b$, and $c$, $a(b + c) = ab

1. **State the problem:** We need to graph the ellipse given by the equation $$16x^2 + 25y^2 + 64x + 50y - 311 = 0$$ and find its center, axes lengths, and intercepts. 2. **Rewrite

1. **State the problem:** A jet travels 4488 miles against the wind in 8 hours and 5368 miles with the wind in 8 hours. We need to find the rate of the jet in still air ($j$) and t

1. **State the problem:** We are given the function $$f(x) = \log_{4} \left( x^{2} - 1 \right)$$ and want to understand its properties and graph shape. 2. **Domain determination:**

1. **State the problem:** Simplify the expression $$\frac{\left(2 \times 1 \times 3 + 3.5\right)}{0.1 - \frac{6}{13}} - 21.7$$ given the numbers and operations. 2. **Rewrite the ex