10. Find a polynomial with zeros \( -2,3 \), and 5 and passes through the point \( (2,24) \)
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To find a polynomial with the given zeros \( -2, 3, \) and \( 5 \), we can express the polynomial in factored form as \( P(x) = k(x + 2)(x - 3)(x - 5) \), where \( k \) is a constant that we will determine using the point \( (2, 24) \). First, plug in \( x = 2 \) and \( P(2) = 24 \): \[ P(2) = k(2 + 2)(2 - 3)(2 - 5) = k(4)(-1)(-3) = 12k. \] Now set this equal to 24: \[ 12k = 24 \implies k = 2. \] So, the polynomial is: \[ P(x) = 2(x + 2)(x - 3)(x - 5). \] Now, expand it: 1. First, expand \( (x + 2)(x - 3) = x^2 - x - 6 \). 2. Then, expand \( (x^2 - x - 6)(x - 5) = x^3 - 5x^2 - x^2 + 5x - 6x + 30 = x^3 - 6x^2 + 30 \). 3. Finally, multiply by 2: \[ P(x) = 2(x^3 - 6x^2 + 30) = 2x^3 - 12x^2 + 60. \] Thus, the polynomial you are looking for is: \[ P(x) = 2x^3 - 12x^2 + 60. \] Now for a bit of fun: Did you know that polynomials are not just numbers on paper? They have played a crucial role in ancient civilizations like the Babylonians, who already knew how to solve quadratic equations over 4000 years ago! Fast-forward to today, and polynomials are fundamental in computer graphics, economics, and even machine learning, allowing us to model everything from curves to complex data patterns! Want to dive deeper into the world of polynomials? Check out "Polynomial Mathematics" by James Stewart, where you can explore polynomial functions, their transformations, and how they apply to real-world problems. You'll find plenty of engaging examples that demystify these mathematical wonders while boosting your problem-solving skills to the next level.