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The equation at hand is x12 + 4x – 1 = 0. To find the solutions, we can analyze the function defined by the left-hand side of the equation. Let’s define: f(x) = x12 + 4x – 1 We want to determine the number of x-values for which f(x) = 0. First, we should investigate … Read more

To understand the relationship between the functions described, we need to break down the equations provided: f(x) = 2g(x) and f(g(x)) = 4g(x). 1. **First Equation Analysis**: The first equation, f(x) = 2g(x), indicates that the function f(x) is defined as twice the value of g(x). This means that for any input x, we can … Read more

When we say that x – 5 is a factor of the polynomial function f(x), we can deduce a crucial piece of information about the function. Specifically, one of the fundamentals of polynomial functions states that if a polynomial f(x) has a factor of the form x – a, then a is a root of … Read more

If a polynomial f(x) has x4 as a factor, there are several important implications regarding the characteristics of the polynomial. Degree of the Polynomial: The degree of the polynomial f(x) must be at least 4. This is because a polynomial of lesser degree cannot have x4 as a factor. If it is a polynomial of … Read more

To determine one of the factors of the given expression x³ – 1331 – 11x² – 11x – 121 – x² – 22x – 121, we first need to simplify it and analyze its polynomial components. The expression can be rewritten for clarity: x³ – (11x² + x²) – (11x + 22x) – (1331 + … Read more

To find the linear approximation of the function f(x) = 4x at a = 0, we start by using the formula for linear approximation, which is given by: L(x) = f(a) + f'(a)(x – a) Where: L(x) is the linear approximation, f(a) is the value of the function at a, f'(a) is the derivative of … Read more

Finding the surface area of a right rectangular prism (also commonly known as a rectangular box) is a straightforward task once you know the dimensions of the prism. The surface area is the total area of all the exterior surfaces of the prism. The formula to calculate the surface area (SA) of a right rectangular … Read more

To estimate the maximum error in the surface area of a sphere using differentials, we start with the measurements provided. The circumference (C) of a sphere can be related to its radius (r) by the formula: C = 2\pi r. 1. **Find the radius using the circumference:** – Using the given circumference: C = 78 … Read more

To find the product of 2i and 5i, follow these steps: Multiply the coefficients: 2 and 5. This gives us: 2 × 5 = 10 Multiply the imaginary units: i × i = i2. By definition, i2 = -1. So we have: i × i = -1 Combine the results: Now you can combine the … Read more

To find one of the factors of the expression 3p5 + 12p3, we first look for common factors in the terms of the expression. 1. **Identify the Coefficients**: The coefficients in the expression are 3 and 12. The greatest common factor (GCF) of 3 and 12 is 3. 2. **Identify the Variables**: The terms also … Read more