answer:The inverse of the given matrix is: left( begin{array}{ccc} frac{5}{13} & frac{4}{13} & frac{4}{13} frac{16}{13} & frac{18}{13} & frac{5}{13} 1 & 0 & 1 end{array} right)

answer:Given the integral: int frac{5x^3-3x^2+2x-1}{x^4+x^2} dx We can simplify this improper fraction by decomposing it into partial fractions. Since the denominator can be factored into: x^4 + x^2 = x^2(x^2 + 1) We have a proper fraction with repeated linear factors and a quadratic factor. We can express the fraction as a sum of simpler fractions: frac{5x^3-3x^2+2x-1}{x^2(x^2 + 1)} = frac{A}{x} + frac{B}{x^2} + frac{Cx + D}{x^2 + 1} Multiplying both sides by the common denominator: 5x^3 - 3x^2 + 2x - 1 = A(x^2 + 1) + Bx^2 + (Cx + D)x^2 We can find the constants A, B, C, and D by comparing coefficients: 1. Setting x = 0 gives us B = -1. 2. Comparing the coefficient of x^3 gives A + C = 5. 3. Comparing the coefficient of x^2 gives -1 + B + D = -3. 4. Comparing the constant term gives A = 2. Now we can solve for the remaining constants: 1. Since A = 2, the equation A + C = 5 becomes 2 + C = 5, so C = 3. 2. Substituting B = -1 and A = 2 into -1 + B + D = -3, we get -1 - 1 + D = -3, so D = -2. The partial fraction decomposition is thus: frac{5x^3-3x^2+2x-1}{x^4+x^2} = frac{2}{x} - frac{1}{x^2} + frac{3x-2}{x^2 + 1} Now we can integrate: int frac{2}{x} - frac{1}{x^2} + frac{3x-2}{x^2 + 1} dx Applying the integral rule for each term: = 2int frac{1}{x} dx - int frac{1}{x^2} dx + int frac{3x}{x^2 + 1} dx - int frac{2}{x^2 + 1} dx = 2ln|x| + frac{1}{x} + frac{3}{2}ln|x^2 + 1| - 2arctan(x) + C Therefore, the result of the indefinite integral is: boxed { bf{ 2ln left|xright| + frac{1}{x} + frac{3}{2}ln left|x^2+1right| - 2arctan left(xright) + C }}

answer:Equating the coefficient of x^3 in the expressions for P(x)P(-x) obtained previously, we have: begin{align} P(x)P(-x) & = -x^6 +x^4(a^2-2b)+cdots P(x)P(-x) & = -(x^3-r^3)(x^3-s^3)(x^3-t^3) & = -x^9+x^3(r^3+s^3+t^3)+cdots end{align} Equating the coefficients of x^3, we get: r^3+s^3+t^3=-b

answer:To find the volume of the cube, we use the formula: {eq}V = s^3 {/eq} where s is the side length of the cube. First, we convert the side length to inches: {eq}15 cm times frac{1 in}{2.54 cm} = 5.91 in {/eq} Then, we convert inches to feet: {eq}5.91 in times frac{1 ft}{12 in} = 0.4925 ft {/eq} Finally, we calculate the volume: {eq}V = (0.4925 ft)^3 = boxed{0.119 ft^3} {/eq}