🧮 algebra

1. **Stating the problem:** We need to analyze and understand the polynomial $$P(x) = (x+2)^2 (x+3)^2 (x-1)^4 (x+1).$$ \n\n2. **Understanding the polynomial:** It's a product of fo

1. Let's first state the problem: Determine if the function $f(x) = x^2$ has an inverse function. 2. To have an inverse, a function must be one-to-one (bijective), meaning each $y$

1. Stating the problem: Given the polynomial $$P(x) = (x + 2)^2 (x + 3)^3 (x - 1)^4 (2x + 1)$$ we need to find its standard form, leading term, leading coefficient, degree, constan

1. We are given two summation expressions and need to verify their correctness. 2. First, consider the summation $$\sum_{r=1}^3 (3r + 4)$$.

1. The problem asks to show that \(\sum_{r=1}^3 (3r + 4) = \sum_{r=1}^3 3r + \sum_{r=1}^3 4\) and \(\sum_{r=1}^4 (4r) = 4 \sum_{r=1}^4 r\). 2. Start with the first equality:

1. Let's state the problem: You asked to solve and explain the math problems in a clear, step-by-step manner. 2. Since you didn't specify the exact problems, I will demonstrate wit

1. The problem asks us to analyze the polynomial \(P(x) = (x+2)^2 (x+3)^2 (x-1)^4 (2x+1)\), which is already factored. 2. First, identify the zeros (roots) of the polynomial by set

1. Stated problem: Simplify the expression $$\frac{3a^5 + a^4 - 3a^3 - 3a^2 + 2}{1 - a^2}$$. 2. Factor the denominator:

1. Find the equation of the line passing through the points $(-3,-5)$ and $(9,1)$. To find the line equation, first calculate the slope $m$:

1. We are given a series with terms from $k=2n$ to $k=n^2$, and there are 9 terms in this series. 2. Recall that the number of terms in a series from $k=a$ to $k=b$ is given by $b

1. **State the problem:** Factorize the expression $n^2 - 2n$. 2. **Identify common factors:** Both terms $n^2$ and $-2n$ have a common factor of $n$.

1. The problem states the equation: $3=3$. 2. This is an identity because both sides of the equation are exactly equal.

1. Problem statement: We are asked to find a function $D(t)$ that describes the temperature difference between a hot cake and the cooler after $t$ minutes. 2. Given: Initial temper

1. **State the problem:** Solve the equation $$x - 7y = 2x + \frac{1}{2}$$ for one variable in terms of the other. 2. **Rewrite the equation:** $$x - 7y = 2x + \frac{1}{2}$$

1. We are given the system of equations to solve for $n$ and $y$: $$2n + y = 3$$

1. We are given the equation $2n + y = 3$. 2. To express $y$ in terms of $n$, subtract $2n$ from both sides:

1. We are given the system of equations: $$y = 14$$

1. **State the problem:** Simplify the expression $$\frac{3a^5+a^4-3a^3-3a^2+2}{1-a^2}$$. 2. **Factor the denominator:** Note that $$1 - a^2$$ is a difference of squares, which fac

1. The problem is to find the value of $A+B$ given variables $A$ and $B$. 2. Without specific values or expressions for $A$ and $B$, we cannot numerically evaluate $A+B$.

1. **State the problem:** Convert the postfix expression $$A B * C D ^ / E F G + * + H I + J K - / -$$ to its equivalent infix form, then evaluate it using values $$A=100, B=10, C=

1. **Problem 8:** The cost of one eraser and three pencils is 17, and the cost of three erasers and four pencils is 31. Find the cost of 5 erasers. 2. Let the cost of one eraser be