🧮 algebra

1. **State the problem:** Expand the expression $ (x + 1)^4 $. 2. **Formula used:** Use the Binomial Theorem which states that $ (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k $

1. **State the problem:** Multiply the polynomials $ (3x - 2) $ and $ (9x^2 + 6x + 4) $. 2. **Formula and rules:** Use the distributive property (also known as the FOIL method for

1. **State the problem:** We need to expand and simplify the expression $$(2z + 5)(4z^2 - 10z + 25)$$.

1. **State the problem:** Simplify the expression $ (y + 1)(y^2 - y + 1) $. 2. **Recall the distributive property:** To multiply two polynomials, multiply each term in the first po

1. **State the problem:** Expand the expression $$(x - 2)(x^2 + 2x + 4)$$. 2. **Formula and rules:** To expand the product of polynomials, use the distributive property (also calle

1. The problem is to find the formula for the expression $ (A+B)^X $. 2. This is a binomial expression raised to a power $X$. The formula used to expand such expressions is called

1. **State the problem:** We are given the linear equation $$y = \frac{4}{3}x - 1$$ and need to understand its slope and y-intercept. 2. **Formula and explanation:** A linear equat

1. The problem asks for the y-intercept of the line given by the equation $$y = \frac{1}{3}x + 4$$. 2. The y-intercept of a line in slope-intercept form $$y = mx + b$$ is the value

1. Let's clarify the problem: You are asking if the same approach applies to a trinomial as it does to other polynomials or expressions. 2. A trinomial is a polynomial with exactly

1. **State the problem:** We want to expand the expression $ (x+1)^3 $. 2. **Formula used:** The cube of a binomial $(a+b)^3$ is expanded using the formula:

1. **State the problem:** We want to find the price per ounce when 48 fluid ounces cost 7.20. 2. **Formula:** Price per ounce = \frac{\text{Total price}}{\text{Total ounces}}

1. **State the problem:** We want to find the price per ounce when 42 fluid ounces cost 7.20. 2. **Formula:** Price per ounce = \frac{\text{Total price}}{\text{Total ounces}}

1. **State the problem:** We need to find two consecutive even integers whose difference of squares is 68. 2. **Define variables:** Let the first even integer be $x$. Since the int

1. The user requests to "Complete on a graph," which is unclear without a specific function or equation. 2. To assist effectively, please provide the function or equation you want

1. We are asked to write the quadratic expression $w^2 + 9w - 5$ in completed square form. 2. The general form of a quadratic is $ax^2 + bx + c$. Here, $a=1$, $b=9$, and $c=-5$.

1. **Problem Statement:** Graph the quadratic functions:

1. **Problem 1:** Write $x^2 + 8x + 5$ in the form $(x + a)^2 + b$ where $a,b$ are integers. 2. **Formula:** To complete the square for $x^2 + bx + c$, use:

1. **Problem statement:** Given that $a$, $b$, $c$, and $d$ are in continued proportion, prove that $$\frac{a-2b}{b-2c} = \frac{3b+4c}{3c+4d}.$$\n\n2. **Recall the definition of co

1. **Stating the problem:** We want to understand the concept of continued proportion, which is a sequence of numbers where each term is in proportion to the next. 2. **Definition:

1. Let's clarify the problem you are referring to and verify the solution. 2. If the problem involves solving an equation or expression where the answer is claimed to be 2, we need

1. **State the problem:** Solve the equation $$-7x - 9 = -2x + 31$$ for $x$. 2. **Write down the equation:** $$-7x - 9 = -2x + 31$$