🧮 algebra

1. The problem asks us to factorise the cubic polynomial $$x^3 + 8x^2 + 17x + 10$$ into linear factors. 2. To factorise, first try to find rational roots using the Rational Root Th

1. State the problem: We need to find which fraction among $\frac{5}{9}$, $\frac{4}{7}$, $\frac{3}{5}$, $\frac{6}{11}$, and $\frac{13}{21}$ has the smallest value. 2. To compare fr

1. The problem asks us to evaluate the sum of fractions: $$\frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} + \frac{1}{110}$$ 2. To add these

1. The problem asks us to find the value of $$\left(4\sqrt{4} + 2\sqrt{3} - \sqrt{49 + 8 \sqrt{3}} \right)^2$$. 2. Start by simplifying the terms inside the parentheses:

1. **State the problem:** Factor the quadratic expression $x^2 + 5x + 6$. 2. **Identify coefficients:** This is a quadratic in standard form $ax^2 + bx + c$, where $a=1$, $b=5$, an

1. Stating the problem: Prove that if $$ (1 + x)^n \geq 1 + nx $$ for $$ x > -1 $$ and $$ n \in \mathbb{N} $$, then $$ (1 + x)^{n+1} \geq 1 + (n+1)x $$. 2. Assume the induction hyp

1. Stating the problem: Solve the system of linear equations: $$-5x + 9y - 7y = 0$$

1. **State the problem:** Find the value of $$\left(4\sqrt{4} + 2\sqrt{3} - \sqrt{49 + 8\sqrt{3}}\right)^2.$$\n\n2. **Simplify the terms inside the parentheses:**\n- Calculate $$4\

1. **State the problem:** We need to find which fraction among $\frac{5}{9}$, $\frac{4}{7}$, $\frac{3}{5}$, $\frac{6}{11}$, and $\frac{13}{21}$ has the smallest value. 2. **Method:

1. We start with the problem: Evaluate \( \frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} + \frac{1}{110} \). 2. First, find a common denomi

1. Problem: For each function, find $f(-x)$ and $-f(-x)$, then compare with $f(x)$ to determine if the function is even, odd, or neither. 2. Function a) $f(x) = x^2 - 4$

1. The **product rule** states that $\log_b(xy) = \log_b x + \log_b y$. This means the log of a product is the sum of the logs. 2. The **quotient rule** states that $\log_b\left(\f

1. Stating the problem: We want to factor the expression $$x^4 + x^2$$. 2. Identify the common factor: Both terms have a factor of $$x^2$$.

1. We are asked to find the domain of each function given. The domain is the set of all real numbers $x$ for which the function is defined. 2. For (a) $f(x) = \sqrt{x+2}$, the expr

1. We are asked to factor each of the given expressions. 2. (a) Factor $4x^2 - 25$.

1. For the pair $f(x) = \frac{1}{x}$ and $g(x) = x$: - Common characteristic: Both are functions that are defined for all $x \neq 0$ (domain excludes zero for $f(x)$), and both are

1. **State the problem:** We need to find the values of $k$ for the quadratic function $$y = kx^2 + (k+3)x - 1$$ such that:\na) The graph cuts the x-axis twice (two distinct real r

1. We are given the half-life of mercury-210 as 600 seconds and an initial mass of 2500 grams. 2. We want to find the remaining mass after 17 minutes.

1. The user requested to use fractions instead of decimals or whole numbers in math problems. 2. Using fractions helps express quantities exactly, avoiding rounding errors common w

1. Let's clarify the problem: You mentioned that for $2b$, 250 isn't a value but 260 is. 2. Assuming you want to find $b$ given a value such as 260 for $2b$, we start with the equa

1. The problem asks us to graph the equation \( y = 2^{x-1} - 3 \). 2. This is an exponential function with base 2, shifted right by 1 unit and down by 3 units.