$qz^9+r$ 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

Understand Factorization: Step Title: Understand the Factorization $\newline$ Concise Step Description: Recognize that if $qz^9 + r$ is a factor of $38z^{18} + bz^9 + 50$ , then for some polynomial $p(z)$ , $38z^{18} + bz^9 + 50 = (qz^9 + r) \cdot p(z)$ . $\newline$ Step Calculation: No specific calculation yet.
Set Up Equation: Step Title: Set Up the Equation $\newline$ Concise Step Description: Assume $p(z) = az^9 + s$ to match the degrees of the polynomial on both sides. $\newline$ Step Calculation: $(qz^9 + r)(az^9 + s) = qaz^{18} + (qs + ar)z^9 + rs$ .
Compare Coefficients: Step Title: Compare Coefficients $\newline$ Concise Step Description: Equate the coefficients from the expanded form to those in $38z^{18} + bz^9 + 50$ . $\newline$ Step Calculation: $qa = 38$ , $qs + ar = b$ , $rs = 50$ .
Solve for $b$ : Step Title: Solve for $b$ $\newline$ Concise Step Description: Focus on solving for $b$ using the equation $qs + ar = b$ . $\newline$ Step Calculation: Since $qa = 38$ , let's choose $q = 1$ and $a = 38$ (simplest case). Then, $b = s + 38r$ .
Maximize $b$ : Step Title: Maximize $b$ $\newline$ Concise Step Description: Maximize $b$ by choosing values for $r$ and $s$ that satisfy $rs = 50$ and maximize $s + 38r$ . $\newline$ Step Calculation: If $r = 1$ , $s = 50$ (since $rs = 50$ ), then $b$ $0$ .
Check Higher Values: Step Title: Check for Higher Values $\newline$ Concise Step Description: Check if larger values of $r$ give a higher $b$ . $\newline$ Step Calculation: If $r = 2$ , $s = 25$ , then $b = 25 + 38 \times 2 = 101$ .
Continue Checking: Step Title: Continue Checking $\newline$ Concise Step Description: Continue to check for even larger values of $r$ . $\newline$ Step Calculation: If $r = 5$ , $s = 10$ , then $b = 10 + 38 \times 5 = 200$ .
Final Check: Step Title: Final Check $\newline$ Concise Step Description: Check if any other values of $r$ and $s$ can maximize $b$ further. $\newline$ Step Calculation: If $r = 10$ , $s = 5$ , then $b = 5 + 38 \times 10 = 385$ .