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Multiplicity in a polynomial refers to the number of times a particular root appears. In other words, if you have a polynomial equation and one of its solutions is a root, the multiplicity indicates how many times that root is counted in the polynomial’s factorization. For example, consider the polynomial P(x) = (x – 2)²(x … Read more

To solve the equation ln(x) + 6ln(2x) = 1, we will start by simplifying the equation using properties of logarithms. Recall that ln(a) + ln(b) = ln(ab). We can apply this to the second term: ln(x) + 6ln(2x) = ln(x) + ln((2x)^6) Now, we can rewrite (2x)^6: (2x)^6 = 2^6 * x^6 = 64x^6. Therefore, … Read more

A triangular prism is a three-dimensional geometric shape that consists of two triangular bases and three rectangular lateral faces. To understand how many vertices a triangular prism has, we can break it down as follows: The prism has two triangular bases, and each triangle has three vertices. This means that the two triangular bases together … Read more

To determine the next number in the sequence 1, 1, 2, 4, 3, 9, 4, we should first analyze the pattern within the numbers. Let’s break it down step-by-step: The first two numbers are 1, 1. The third number is 2, which can be seen as the sum of the first two numbers: 1 + … Read more

To determine the equation of a plane when you have its x, y, and z intercepts, you can use the intercept form of the equation of a plane. The intercepts are as follows: X-intercept (a): This is the point where the plane intersects the x-axis. In this case, it is 2, which means the plane … Read more

The volume V of a sphere is given by the formula: V = (4/3)πr³ where r is the radius of the sphere, and π (pi) is a constant approximately equal to 3.14159. To find the rate of change of the volume with respect to the radius, we need to calculate the derivative of the volume … Read more

A kilometer is a unit of length in the metric system that is equal to 1,000 meters. This means that if you have a distance of 1 kilometer, it can be converted to meters simply by multiplying by 1,000. To give you a better understanding, imagine walking or running a distance of 1 kilometer. It … Read more

To simplify the expression 7a + 2b + 5a + b, we first need to combine like terms. Like terms are terms that have the same variable raised to the same power. 1. **Identify like terms**: In our expression, we have: 7a and 5a, which are both terms with the variable a. 2b and b … Read more

The mode is defined as the value that appears most frequently in a dataset. In statistical terms, it represents the number(s) with the highest frequency. When it comes to a dataset where all the numbers are different, understanding the mode can be a bit perplexing. Since each number occurs only once, there is no number … Read more

To convert the quadratic function f(x) = x² + 8x + 3 into vertex form, we need to complete the square. Vertex form is typically expressed as: f(x) = a(x – h)² + k where (h, k) is the vertex of the parabola represented by the quadratic function. Here are the steps to achieve this: … Read more