🧮 algebra

1. Masalah: Hitung determinan matriks $T$ dan $D$ menggunakan metode kofaktor dengan memilih baris atau kolom yang mudah. 2. Rumus determinan dengan kofaktor: $$\det(A) = \sum_{j=1

1. Masalah: Buktikan bahwa jika sistem persamaan linear $Ax=b$ memiliki lebih dari satu solusi, maka sistem tersebut memiliki tak hingga banyak solusi. 2. Misalkan $x_1$ dan $x_2$

1. **State the problem:** Ignacio's legs start at 49.3 cm from the floor, and the desk bottom is at 74.5 cm. Each clockwise rotation raises his legs by 4.8 cm. We want to find the

1. **State the problem:** Determine whether the function $f(x) = x^2$ is one-to-one (injective) when the domain and codomain are both the set of integers. 2. **Recall the definitio

1. **Problem Statement:** Determine whether the function $f(x) = b x^2$ is one-to-one (injective) when the domain and codomain are both the set of integers. 2. **Definition of One-

1. Masalah yang dibahas adalah mengapa pembuktian Teori 1.6.1 hanya menunjukkan kasus ketika sistem linear memiliki banyak solusi (infinitely many solutions). 2. Teori 1.6.1 menyat

1. Diketahui polinomial $p(x) = ax^5 + bx - 1$ dengan $a$ dan $b$ konstan. 2. Jika $p(x)$ dibagi oleh $(x - 2.006)$, sisanya adalah 3, maka menurut Teorema Sisa:

1. **State the problem:** Simplify the expression $$\left\{\left[\left(\frac{2}{3}\right)^{-4}\right]^{\frac{2}{3}} \cdot 1.5 \cdot \left(\frac{3}{2}\right)^{\frac{1}{9}} \cdot 121

1. Masalah ini meminta kita menghitung nilai dari ekspresi postfix: 6 2 / 4 + 3 *. 2. Ekspresi postfix (atau notasi Polandia terbalik) berarti operator mengikuti operandnya.

1. **Problem:** Calculate $$\left[\left(\frac{2}{3}\right)^{-4}\right]^{\frac{2}{3}}$$ and $$-1.5 : \left(\frac{3}{2}\right)^{\frac{1}{9}} \cdot 121^{\frac{5}{8}}$$. **Step 1:** Si

1. **Problem statement:** Expand and reduce the polynomial $$Q(x) = (4x + 5)^2 - (x - 2)(4x + 5)$$. 2. **Recall formulas:**

1. The problem asks: For the function $f(x) = -5$, where is it negative? 2. Since $f(x) = -5$ is a constant function always equal to $-5$, it is negative for all real numbers.

1. The problem is to solve the inequality involving $x$ where the solution is given as $x<2$ or $x>3$. 2. This type of inequality typically arises from expressions like $(x-2)(x-3)

1. **Stating the problem:** We want to decompose the function $$y(p) = \frac{1}{p^2 + 4} + \frac{1 - e^{-p}}{p(p^2 + 4)}$$ into partial fractions. 2. **Rewrite the expression:** Co

1. **Problem Statement:** Verify if the functions $f(x) = \frac{1}{2}x - 7$ and $g(x) = 2x - 14$ are inverse functions by using composition of functions. 2. **Formula and Rule:** T

1. Let's start by stating the problem: A quadratic inequality involves an expression of the form $ax^2 + bx + c$ where $a \neq 0$, and we want to find the values of $x$ that make t

1. The problem asks for the "primary capacity" of the complex number $\cos(50) + i \sin(50) + i$. 2. First, recognize that $\cos(50) + i \sin(50)$ is in the form of Euler's formula

1. נניח את הבעיה: נתונה הפונקציה $$h(x) = \frac{x^2 - 1}{x^2 + 1}$$ על התחום $$(-\infty, \infty)$$. 2. נבדוק את תחום ההגדרה: המונה הוא $$x^2 - 1$$ והמכנה הוא $$x^2 + 1$$. מאחר ש-$$

1. ננתח כל טענה בנפרד כדי לבדוק אם היא נכונה או לא. 2. טענה ראשונה: אם $x \notin \mathbb{Q}$ ו-$y \in \mathbb{Q}$ אז בהכרח $x + y \notin \mathbb{Q}$.

1. **State the problem:** We need to divide the polynomial $$x^4 - 8x^3 + (5a - 1)x^2 + 6x - 3a - 6$$ by $$x^2 - 1$$.

1. **Stating the problem:** Multiply the polynomials $$ (x^4 - 8x^3 + (5a - 1)x^2 + 8x - 3a - 6) \cdot (x^2 - 1) $$.