= z^3 + 8z^2 + 21.333z + 18.96 - 8z^2 - \frac{128}{3}z - \frac{512}{9} + 9z + 6

["# Simplifying and Factoring: A Comprehensive Solution to the Polynomial Expression ( z^3 + 8z^2 + 21.333z + 18.96 - 8z^2 - \frac{128}{3}z - \frac{512}{9} + 9z + 6 )", "Polynomial expressions can appear daunting at first glance, especially when dealing with mixed coefficients and fractional terms. However, simplifying and factoring these expressions step by step unlocks clarity and reveals their core structure. In this article, we rigorously simplify and factor the expression:", "[\nP(z) = z^3 + 8z^2 + 21.333z + 18.96 - 8z^2 - \frac{128}{3}z - \frac{512}{9} + 9z + 6\n]", "But before we factor, let’s simplify this expression into a clean standard cubic polynomial.", "---", "## Step 1: Simplify the Expression", "Start by combining like terms:", "### Combine ( z^2 ) terms:\n[\n8z^2 - 8z^2 = 0\n]", "### Combine ( z ) terms:\nConvert 21.333 to a fraction for easier arithmetic. Note that ( 21.333 \approx \frac{64}{3} ), since ( \frac{64}{3} \approx 21.333\overline{3} ). For high precision and consistency with fraction coefficients below, we use:", "[\n21.333 \approx \frac{64}{3}\n]", "So,\n[\n\frac{64}{3}z - \frac{128}{3}z + 9z = \left(\frac{64}{3} - \frac{128}{3} + 9\right)z\n= \left(-\frac{64}{3} + 9\right)z\n= \left(-\frac{64}{3} + \frac{27}{3}\right)z\n= -\frac{37}{3}z\n]", "### Combine constant terms:\n[\n18.96 \approx \frac{1704}{90} \quad \ ext{(but better: use fractions consistent with others)}\n]", "But note:\n( 18.96 = \frac{1896}{100} = \frac{474}{25} ), and\n( -\frac{512}{9} ) and ( +6 = \frac{54}{9} )", "Instead, better to convert all decimals to fractions:", "- ( 21.333 = \frac{64}{3} )\n- ( 18.96 = \frac{1896}{100} = \frac{474}{25} )\n- ( -\frac{128}{3} ) remains as is\n- ( -\frac{512}{9} )\n- ( 9z = \frac{27}{3}z ), but we already combined", "So simplifying the full expression:", "[\nP(z) = z^3 + \left( \frac{64}{3}z - \frac{128}{3}z + 9z \right) + \left( \frac{474}{25} + 6 - \frac{512}{9} \right)\n]", "Simplify coefficients:", "Z-terms:\n[\n\frac{64 - 128 + 27}{3}z = \frac{-37}{3}z\n]", "Constants:\n[\n\frac{474}{25} + \frac{150}{25} - \frac{512}{9} = \frac{624}{25} - \frac{512}{9}\n]", "Find common denominator for ( \frac{624}{25} - \frac{512}{9} ):", "LCM of 25 and 9 is ( 225 ):", "[\n\frac{624}{25} = \frac{624 \ imes 9}{225} = \frac{5616}{225}, \quad \frac{512}{9} = \frac{512 \ imes 25}{225} = \frac{12800}{225}\n]", "So,\n[\n\frac{5616 - 12800}{225} = \frac{-7192}{225}\n]", "And ( 6 = \frac{54}{9} ), already converted to 25 denominator:\nBut wait — we already expressed constants in 25: actual constant is ( \frac{474 + 150}{25} - \frac{512}{9} = \frac{624}{25} - \frac{512}{9} ), already computed as ( -\frac{7192}{225} )", "Now convert entire constant to denominator 225:", "[\n\frac{474}{25} = \frac{4242}{225}, \quad \frac{512}{9} = \frac{12800}{225}\n\Rightarrow \frac{4242 - 12800}{225} = \frac{-8558}{225}\n]", "Now with denominator 225:", "[\nP(z) = z^3 - \frac{37}{3}z - \frac{8558}{225}\n]", "But this is complicated — perhaps use decimal approximations for clarity in factoring?", "Alternatively, notice the original expression may have been constructed to factor nicely.", "Let’s revisit the expression with exact fractions:", "Given:\n[\nP(z) = z^3 + 8z^2 + \left( \frac{64}{3} - \frac{128}{3} + 9 \right)z + \left( 18.96 - \frac{512}{9} + 6 \right)\n]", "We already computed:", "- ( z )-coefficient: ( \frac{64 - 128 + 27}{3} = \frac{-37}{3} )\n- Constant term:\n ( 18.96 = \frac{474}{25} ), ( 6 = \frac{150}{25} ), so ( \frac{474 + 150}{25} = \frac{624}{25} )\n Then ( \frac{624}{25} - \frac{512}{9} = \frac{5616 - 12800}{225} = \frac{-7184}{225} )? Wait — earlier miscalculation?", "Wait: earlier we had ( 18.96 \approx \frac{474}{25} = 18.96 ), yes.", "But ( \frac{474}{25} - \frac{512}{9} = \frac{4266 - 6400}{225} = \frac{-2134}{225} )? Retry:", "( \frac{474}{25} = \frac{474 \ imes 9}{225} = \frac{4266}{225} ),\n( \frac{512}{9} = \frac{512 \ imes 25}{225} = \frac{12800}{225} ),\nSo:\n[\n\frac{4266 - 12800}{225} = \frac{-8534}{225}\n]", "Wait — consistent calculation:\nActually:", "[\n\frac{474}{25} - \frac{512}{9} = \frac{474 \cdot 9 - 512 \cdot 25}{225} = \frac{4266 - 12800}{225} = \frac{-8534}{225}\n]", "Then add 6:\n( 6 = \frac{1350}{225} ), so total:", "[\n\frac{-8534 + 1350}{225} = \frac{-7184}{225}\n]", "Now:\n[\nP(z) = z^3 - \frac{37}{3}z - \frac{7184}{225}\n]", "Still messy. Instead, consider original expression may have been meant to simplify to something like:", "Try factoring heuristically: suppose ( P(z) = z^3 - \frac{37}{3}z - \frac{7184}{225} )", "But this suggests a rational root by Rational Root Theorem? Try rational values.", "Alternatively, let’s suppose a typo or interpretation issue — perhaps the original expression was:", "[\nz^3 + 8z^2 + 21.333z + 18.96 - 8z^2 - \frac{128}{3}z - \frac{512}{9} + 9z + 6\n]", "Group perfectly:", "[\n= z^3 + (8 - 8)z^2 + \left(21.333 - \frac{128}{3} + 9 + 6\right)z + \left(18.96 - \frac{512}{9} + 6\right)\n]", "Convert all to fractions:", "- ( 21.333 = \frac{64}{3} )\n- ( 9 + 6 = 15 )\n- So z-coefficient: ( \frac{64}{3} - \frac{128}{3} + 15 = -\frac{64}{3} + 15 = -\frac{64}{3} + \frac{45}{3} = -\frac{19}{3} )", "Wait — earlier inconsistency: previously added ( 9z = \frac{27}{3}z ), but here total constant is 15, not matched.", "Ah — mistake: the ( +9z ) was already included in coefficient sum?", "No — we must combine all ( z )-terms:", "Original:\n[\n\frac{64}{3}z"]