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Finding All Zeros of the Polynomial To find all the zeros of the polynomial function f(x) = x^4 + 45x^2 + 196, we start by recognizing that we already know one of the zeros is 2i (a complex number). Since the coefficients of our polynomial are real, the complex zeros will occur in conjugate pairs. … Read more

The expression you are looking to analyze is the product of three terms: x, 5x, and 2. To find the middle term, we will first multiply these terms together, and then analyze the resulting polynomial. Let’s start by multiplying the terms: Multiply x and 5x: x * 5x = 5x^2 Next, multiply the result by … Read more

To find out how many different groups of 3 employees can be formed from a total of 5 employees, we can use the concept of combinations. Combinations are used when the order of selection does not matter, which is the case here since we are simply selecting a group. The formula for combinations is given … Read more

To find the perimeter of a rhombus when you know the lengths of the diagonals, you can use the following steps: Determine the Lengths of the Diagonals: In this case, the lengths of the diagonals are given as 10 cm and 24 cm. Find the Length of One Side: The diagonals of a rhombus bisect … Read more

General Solution for the System of Differential Equations To find the general solution of the system: dx/dt = 7x – y dy/dt = 5x – 3y We start by rewriting the system in matrix form: d/dt X = AX where X = [ x y ], A = [ 7 -1 ] [ 5 -3 … Read more

Understanding Polynomial Division When you divide a polynomial by a linear expression, you can gain insights into the behavior of the polynomial, such as its roots and factors. In this case, we’re looking at the polynomial f(x) = 4x³ + 3x² + ax + b and dividing it by the factors x – 1 and … Read more

Finding f(x) over g(x) To find the value of f(x) divided by g(x), we first need to define the functions: f(x) = 2×3 + x2 – 16x – 15 g(x) = x – 3 We will calculate f(x) and g(x) separately and then perform the division. Step 1: Calculate f(x) Let’s find f(x): For simplicity, … Read more

To solve for the value of Cos(α) given that Sin(α) = 1/4 and Tan(α) = 0, we can follow these steps: Understanding Tan(α): The tangent of an angle α is defined as the ratio of the sine to cosine: Tan(α) = Sin(α) / Cos(α) If Tan(α) = 0, it implies that Sin(α) = 0 or … Read more

To find the roots of the quadratic equation y = x² + 3x + 10, we will use the quadratic formula. The quadratic formula is given by: x = (-b ± √(b² – 4ac)) / 2a In this equation, a, b, and c are the coefficients from the standard form of a quadratic equation ax² … Read more

To approximate the real number solutions of the polynomial function f(x) = x³ – 2x² + 5x – 6, we can use several methods, including graphing, the intermediate value theorem, or numerical methods like the Newton-Raphson method. Here’s a step-by-step guide: Step 1: Analyze the Function First, let’s evaluate the function at a few points … Read more