🧮 algebra

1. **Problem 1:** Given $f(x) = \frac{(1 + 2x)^2}{(1 - 3x)^2}$ (i) Find the first 4 terms in the power series expansion.

1. **State the problem:** Solve the equation $$\frac{n \times n + n}{n} = 10$$ for $n$. 2. **Rewrite the equation:** The numerator is $n \times n + n = n^2 + n$, so the equation be

1. **State the problem:** Solve the equation $n \times n + \frac{n}{n} = 10$ for $n$. 2. **Rewrite the equation:** The equation is $n^2 + \frac{n}{n} = 10$.

1. **State the problem:** Solve the equation $$\frac{\lambda n + n}{\lambda} = 10$$ for $n$ in terms of $\lambda$. 2. **Rewrite the equation:** The numerator is $\lambda n + n$, wh

1. **Problem:** Solve the linear equation $\frac{2x + 5}{3} = 11$. Multiply both sides by 3 to eliminate the denominator:

1. **State the problem:** Factor the quadratic expression $x^2 + 5x + 6$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multipl

1. **State the problem:** Simplify the expression $$\frac{n \times n + n}{n}$$ when $$n = 10$$. 2. **Write the expression:** The expression is $$\frac{n^2 + n}{n}$$ because $$n \ti

1. **Problem 24:** Identify the graph of the equation $y=\frac{1}{3}x - 1$. 2. The equation is in slope-intercept form $y=mx+b$ where $m=\frac{1}{3}$ (positive slope) and $b=-1$ (y

1. **State the problem:** Express $$\frac{3 \left(e^{j400} + e^{-2j00}\right)}{e^{j00}}$$ in terms of cosine only.

1. The problem is to solve the equation given in question 4 (please provide the exact equation if needed). 2. Generally, to solve algebraic equations, we use the principle of isola

1. State the problem. Use the Factor Theorem to determine whether $x-2$ is a factor of $f(x)=x^3-4x^2+x+6$, and to factor $f(x)$ completely.

1. The Factor Theorem states that if a polynomial $f(x)$ has a factor $(x - a)$, then $f(a) = 0$. 2. To use the Factor Theorem, substitute $x = a$ into the polynomial and check if

1. **State the problem:** We are given the slope formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ as $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ and we need to solve for $y_2$. 2.

1. The problem is to understand the formula for the slope of a line given two points $(x_1, y_1)$ and $(x_2, y_2)$. 2. The slope formula is given by $$m = \frac{y_2 - y_1}{x_2 - x_

1. The problem asks to find the coefficients of certain powers of $x$ in binomial expansions. 2. Recall the binomial theorem: $$(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$$

1. **State the problem:** We need to add the fractions $\frac{2}{6}$ and $\frac{14}{13}$.\n\n2. **Formula and rules:** To add fractions, they must have a common denominator. The fo

1. We are asked to add the fractions $\frac{4}{5} + \frac{16}{14}$.\n 2. To add fractions, we need a common denominator. The denominators are 5 and 14. The least common denominator

1. We are asked to solve and simplify the following fraction addition problems: 2. $\frac{3}{9} + \frac{5}{2} = 2 \frac{5}{6}$

1. We are asked to find the constant term in the expansion of $\left(4x^2 - \frac{1}{2x}\right)^6$. 2. The binomial expansion formula is:

1. **State the problem:** We are given two numbers where their sum is 59, and the smaller number is 21 less than the larger number. 2. **Define variables:** Let the larger number b

1. **State the problem:** Solve the system of linear equations: $$5x + 6y = -5$$