🧮 algebra

1. **Problem statement:** Simplify the polynomial expressions: i. $$\frac{2z^3 + 5z + 8}{z + 1}$$

1. **Find the domain of the function** $f(x) = \frac{1}{x^2 - 9}$. 2. The domain of a function is the set of all $x$ values for which the function is defined.

1. **Problem statement:** Find the slope of a line passing through the origin and the midpoint of the segment joining points $P(0,-4)$ and $B(8,0)$. 2. **Formula for midpoint:** Th

1. **Stating the problem:** The grade 6 volleyball team scored five more than twice as many points as its opponents. We need to write the correct equation representing this situati

1. **State the problem:** Ivan's weight is 4 kg more than Ethan's weight.

1. **Problem:** Suppose $x$ is a rational number and $y$ is an irrational number. Determine which of the following statements is necessarily true: A. $xy$ is rational.

1. **Problem statement:** A shopkeeper marks a shirt at 1200 and gives a 20% discount, yet earns a profit of 25. We need to find the real price (cost price) of the shirt. 2. **Form

1. The problem is to simplify or understand the expression $\sqrt[5]{2}$, which is the fifth root of 2. 2. The fifth root of a number $a$ is the number $x$ such that $x^5 = a$. In

1. The problem is to simplify or understand the expression $\sqrt[b]{2}$, which is the $b$-th root of 2. 2. The $b$-th root of a number $a$ is defined as the number which, when rai

1. **Stating the problem:** We are given the function $$z = \frac{\sqrt[3]{x}}{y}$$ and another function $$f(x) = y$$. 2. **Understanding the functions:** Here, $$z$$ depends on bo

1. **State the problem:** Solve the equation $$\frac{1}{5}(x-8) + 4 + \frac{x}{7} = -23 - \frac{x}{5}$$ for $x$. 2. **Identify the formula and rules:** To solve linear equations wi

1. **State the problem:** Solve the equation $x^2 - x^3 = 80$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. **State the problem:** Solve the equation $$\frac{4(x+2)}{3} - \frac{6(x-7)}{7} = 12$$ for $x$. 2. **Formula and rules:** To solve equations with fractions, first find a common

1. **Problem statement:** Find the slope of a line that makes a 30° angle with the positive direction of the y-axis measured anticlockwise. 2. **Understanding the problem:** The sl

1. **State the problem:** Solve the equation $$\frac{4(x+2)}{3} - \frac{6(x-)}{7} = 12$$. Note there seems to be a typo in the second term: \(6(X-)\) is incomplete. Assuming it sho

1. The problem asks whether the expression $$\lfloor x + y \rfloor$$ can be greater than $$\lfloor x \rfloor + \lfloor y \rfloor$$. 2. Recall the definition of the floor function:

1. The problem asks for the definition of the floor function of a real number $x$. 2. The floor function, denoted as $\lfloor x \rfloor$, is defined as the largest integer less tha

1. **State the problem:** Factorize the expression $$4x^2 - \frac{1}{16}$$. 2. **Recognize the form:** This is a difference of squares, which follows the formula $$a^2 - b^2 = (a -

1. **State the problem:** Simplify the expression $$\frac{7a - 1}{2a^2 + 6a} + \frac{5 - 3a}{a^2 - 9}$$. 2. **Identify the denominators and factor them:**

1. **State the problem:** Simplify the expression $$\frac{7a - 1}{2a^2 + 6a} + \frac{5 - 3a}{a^7 - 8}$$. 2. **Analyze each denominator:**

1. **State the problem:** Calculate the value of $5^{16}$. 2. **Formula used:** The expression $a^n$ means multiplying the base $a$ by itself $n$ times.