answer:The factored form of the polynomial is 4 (1-x) (5-x) (x-7).

answer:To find the dot product of vectors vec{u} and vec{v}, we multiply corresponding components and sum them: vec{u} cdot vec{v} = left(frac{47}{7}right) left(frac{8}{7}right) + left(-frac{38}{7}right) left(frac{64}{7}right) + left(-frac{32}{7}right) left(frac{2}{7}right) - left(frac{23}{7}right) left(-frac{5}{7}right) + left(frac{18}{7}right) (10) - left(-frac{29}{7}right) left(frac{38}{7}right) - left(-frac{66}{7}right) left(-frac{2}{7}right) + left(frac{44}{7}right) left(-frac{40}{7}right) Now, let's calculate each term: begin{align*} &frac{47cdot8}{7^2} - frac{38cdot64}{7^2} - frac{32cdot2}{7^2} + frac{23cdot5}{7^2} + frac{18cdot10}{7} - frac{29cdot38}{7^2} + frac{66cdot2}{7^2} - frac{44cdot40}{7^2} &= frac{376}{49} - frac{2432}{49} - frac{64}{49} + frac{115}{49} + frac{180cdot7}{7^2} - frac{1102}{49} + frac{132}{49} - frac{1760}{49} &= frac{376 - 2432 - 64 + 115 + 1260 - 1102 + 132 - 1760}{49} &= frac{-3475}{49} end{align*} Therefore, the dot product is: vec{u} cdot vec{v} = frac{-3475}{49}

answer:Let y = x+frac{1}{x+frac{1}{x+ddots}}. Then, we have y = x+frac{1}{y}. Solving for y, we get y^2 - xy - 1 = 0. Substituting x = frac{992}{1923}, we get y^2 - frac{992}{1923}y - 1 = 0. Using the quadratic formula, we find that y = frac{992 pm sqrt{992^2 + 4 cdot 1923}}{2 cdot 1923}. Simplifying, we get y = frac{992 pm sqrt{3943945}}{1923}. Since y is positive, we have y = frac{992 + sqrt{3943945}}{1923}. Therefore, frac{1}{x+frac{1}{x+frac{1}{x+ddots}}} = frac{1}{y} = frac{1923}{992 + sqrt{3943945}}. Rationalizing the denominator, we get frac{1}{x+frac{1}{x+frac{1}{x+ddots}}} = frac{1923}{992 + sqrt{3943945}} cdot frac{992 - sqrt{3943945}}{992 - sqrt{3943945}}. Simplifying, we get frac{1}{x+frac{1}{x+frac{1}{x+ddots}}} = frac{1923(992 - sqrt{3943945})}{992^2 - 3943945}. Expanding and simplifying, we get frac{1}{x+frac{1}{x+frac{1}{x+ddots}}} = frac{1923(992 - sqrt{3943945})}{3943945 - 992^2}. Finally, we get frac{1}{x+frac{1}{x+frac{1}{x+ddots}}} = frac{sqrt{3943945}-496}{1923}. The answer is frac{sqrt{3943945}-496}{1923}

answer:A consumer can maximize their utility by allocating their budget in a way that equalizes the marginal utility per dollar spent on each product. This means that for every additional dollar spent on a product, the consumer should experience an equal increase in satisfaction. If the marginal utility per dollar is higher for one product than another, the consumer should purchase more of the product with the higher marginal utility and less of the other until the marginal utilities per dollar are equalized. By doing so, the consumer ensures that they are getting the most satisfaction possible from their available budget.