qz^9+r is a factor of 38z^18 + bz^9 + 50 What is the highest possible value of b?

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qz^9+r is a factor of 38z^18 + bz^9 + 50  What is the highest possible value of b?

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Understand Factorization: Step Title: Understand the Factorization Concise Step Description: Recognize that if qz^9 + r is a factor of 38z^18 + bz^9 + 50, then for some polynomial p(z), 38z^18 + bz^9 + 50 = (qz^9 + r) * p(z). Step Calculation: No specific calculation yet.Set Up Equation: Step Title: Set Up the Equation Concise Step Description: Assume p(z) = az^9 + s to match the degrees of the polynomial on both sides. Step Calculation: (qz^9 + r)(az^9 + s) = qaz^18 + (qs + ar)z^9 + rs.Compare Coefficients: Step Title: Compare Coefficients Concise Step Description: Equate the coefficients from the expanded form to those in 38z^18 + bz^9 + 50. Step Calculation: qa = 38, qs + ar = b, rs = 50.Solve for b: Step Title: Solve for b Concise Step Description: Focus on solving for b using the equation qs + ar = b. Step Calculation: Since qa = 38, let's choose q = 1 and a = 38 (simplest case). Then, b = s + 38r.Maximize b: Step Title: Maximize b Concise Step Description: Maximize b by choosing values for r and s that satisfy rs = 50 and maximize s + 38r. Step Calculation: If r = 1, s = 50 (since rs = 50), then b = 50 + 38*1 = 88.Check Higher Values: Step Title: Check for Higher Values Concise Step Description: Check if larger values of r give a higher b. Step Calculation: If r = 2, s = 25, then b = 25 + 38*2 = 101.Continue Checking: Step Title: Continue Checking Concise Step Description: Continue to check for even larger values of r. Step Calculation: If r = 5, s = 10, then b = 10 + 38*5 = 200.Final Check: Step Title: Final Check Concise Step Description: Check if any other values of r and s can maximize b further. Step Calculation: If r = 10, s = 5, then b = 5 + 38*10 = 385.

$qz^9+r$ is a factor of $38z^{18} + bz^9 + 50$ . What is the highest possible value of $b$ ?

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qz9+rqz^9+rqz9+r is a factor of 38z18+bz9+5038z^{18} + bz^9 + 5038z18+bz9+50. What is the highest possible value of bbb?

$qz^9+r$ is a factor of $38z^{18} + bz^9 + 50$ . What is the highest possible value of $b$ ?