To find the remainder when the polynomial x³ + 4x² + 12x + 9 is divided by x², we can employ the polynomial long division method. 1. **Setup the Division**: Start by setting up the polynomial division. We want to divide x³ + 4x² + 12x + 9 by x². 2. **Perform the Division**: When … Read more
To find the distance between two points in a Cartesian coordinate system, we can apply the distance formula. The distance d = √((x2 – x1)2 + (y2 – y1)2) where (x1, y1) and (x2, y2) are the coordinates of the two points. We have the … Read more
To simplify the expression 3 + 4i + 5 + 2i, we start by combining like terms. This expression consists of real numbers and imaginary numbers: Real parts: 3 and 5 Imaginary parts: 4i and 2i Next, add the real parts together: 3 + 5 = 8 Then, add the imaginary parts: 4i + 2i … Read more
The point-slope equation that you provided is y = 3x + 5. To clarify how we derive the linear function from this point-slope form, let’s break it down step by step. The point-slope form of a linear equation is generally expressed as:y – y1 = m(x – x1) In this equation:– m represents the slope … Read more
To solve the linear inequality y < 4x + 5, we will follow a series of steps to find the solution set. 1. Understanding the inequality: The inequality states that for any point (x, y) that satisfies the condition, the y-value must be less than the value given by the equation of the line y … Read more
To solve the problem of finding the smallest value of the sum of the squares of two positive numbers whose total is 16, let’s denote the two numbers as x and y. Given the constraint: x + y = 16 We want to minimize the sum of their squares: S = x² + y² We … Read more
The magnitude of the cross product of two vectors a and b, denoted as |a × b|, is a measure of the area of the parallelogram formed by these two vectors. This geometric interpretation is very useful in understanding the relationship between the vectors in three-dimensional space. The formula to calculate the magnitude of the … Read more
Examples of Cone, Sphere, Cylinder, and Cuboid Understanding the shapes of geometric solids is essential in various fields such as architecture, engineering, and art. Below are everyday examples of cones, spheres, cylinders, and cuboids: Cone Examples Traffic Cones: These are used to direct traffic and ensure safety in construction areas. Ice Cream Cones: A delicious … Read more
Finding Two Numbers To solve the problem where the sum of two numbers is 21 and their difference is 5, we can use a system of equations. Let’s denote the two numbers as x and y. We can set up the following equations based on the information given: Equation 1: x + y = 21 … Read more
The equation 3x + 2y = 4 is a linear equation in two variables, x and y. To understand what the graph of this equation represents, let’s break it down step by step. The first step in exploring the graph is to rewrite the equation in slope-intercept form, which is y = mx + b, … Read more