This biology question covers important biological concepts and processes. The step-by-step explanation below helps you understand the underlying mechanisms and reasoning.
ScanSolve AI Answer
Step 1: Apply the Factor Theorem. The Factor Theorem states that if (ax+b) is a factor of a polynomial P(x), then P(-(b)/(a)) = 0. Given that (3x+2) is a factor of P(x) = 18x^3 + bx^2 - x - 2, we set 3x+2=0 to find the root. 3x = -2 x = -(2)/(3) According to the Factor Theorem, P(-(2)/(3)) must be equal to 0. Step 2: Substitute x = -(2)/(3) into P(x) and set the expression to zero. P(-(2)/(3)) = 18(-(2)/(3))^3 + b(-(2)/(3))^2 - (-(2)/(3)) - 2 = 0 Step 3: Simplify the expression. Calculate the powers: (-(2)/(3))^3 = -(2^3)/(3^3) = -(8)/(27) (-(2)/(3))^2 = (2^2)/(3^2) = (4)/(9) Substitute these values back into the equation: 18(-(8)/(27)) + b((4)/(9)) - (-(2)/(3)) - 2 = 0 -(18 · 8)/(27) + (4b)/(9) + (2)/(3) - 2 = 0 Simplify the first term: (18)/(27) = (2 · 9)/(3 · 9) = (2)/(3). So, -(2 · 8)/(3) = -(16)/(3). The equation becomes: -(16)/(3) + (4b)/(9) + (2)/(3) - 2 = 0 Step 4: Solve for b. Combine the constant terms: (-(16)/(3) + (2)/(3)) + (4b)/(9) - 2 = 0 -(14)/(3) + (4b)/(9) - 2 = 0 To eliminate the denominators, multiply the entire equation by the least common multiple of 3 and 9, which is 9: 9(-(14)/(3)) + 9((4b)/(9)) - 9(2) = 9(0) -3 · 14 + 4b - 18 = 0 -42 + 4b - 18 = 0 Combine the constant terms: 4b - 60 = 0 Add 60 to both sides: 4b = 60 Divide by 4: b = (60)/(4) b = 15 The value of b is 15. 3 done, 2 left today. You're making progress.