\( f(w) = w^3 - 8w^2 + 9w - 18 \), also ohne \( w^2 \) bei Transformation? Besser: mache Substitution \( w = z + \frac{8}{3} \).
["Transform and Simplify: Analyzing the Function ( f(w) = w^3 - 8w^2 + 9w - 18 ) via a Strategic Linear Substitution", "When studying cubic functions like ( f(w) = w^3 - 8w^2 + 9w - 18 ), direct analysis—such as computing derivatives, locating extrema, or finding roots—can become algebraically cumbersome. A powerful transformation preserves the function’s essential shape while simplifying key aspects, making it ideal for deeper mathematical insight. This article explores how to simplify ( f(w) ) using a clever substitution: ( w = z + \frac{8}{3} ), eliminating the quadratic term and enabling a cleaner analysis.", "### Why Transform ( f(w) )?", "Cubic polynomials often feature a quadratic term that complicates vertex finding, symmetry analysis, and root computation. Removing this term refines the functional form, revealing clearer structural properties. The substitution ( w = z + \frac{8}{3} ) is inspired by the vertex-like transformation that centers the function around the axis of symmetry, typically located at the negative coefficient of ( w^2 ) divided by 3.", "### Step 1: Substitute ( w = z + \frac{8}{3} )", "Let ( w = z + \frac{8}{3} ). We substitute this into ( f(w) ):", "[\nf(z + \ frac{8}{3}) = \left(z + \ frac{8}{3}\right)^3 - 8\left(z + \ frac{8}{3}\right)^2 + 9\left(z + \ frac{8}{3}\right) - 18\n]", "We now expand each term carefully.", "#### Expand ( \left(z + \frac{8}{3}\right)^3 )", "[\n= z^3 + 3z^2 \cdot \frac{8}{3} + 3z \left(\frac{8}{3}\right)^2 + \left(\frac{8}{3}\right)^3\n= z^3 + 8z^2 + 3z \cdot \frac{64}{9} + \frac{512}{27}\n= z^3 + 8z^2 + \frac{192}{9}z + \frac{512}{27}\n= z^3 + 8z^2 + \frac{64}{3}z + \frac{512}{27}\n]", "#### Expand ( -8\left(z + \frac{8}{3}\right)^2 )", "[\n= -8\left(z^2 + 2z \cdot \frac{8}{3} + \left(\frac{8}{3}\right)^2\right)\n= -8\left(z^2 + \frac{16}{3}z + \frac{64}{9}\right)\n= -8z^2 - \frac{128}{3}z - \frac{512}{9}\n]", "#### Expand ( 9\left(z + \frac{8}{3}\right) )", "[\n= 9z + 9 \cdot \frac{8}{3} = 9z + 24\n]", "### Step 2: Combine All Terms", "Add all the expanded expressions:", "[\nf(z + \ frac{8}{3}) = \left(z^3 + 8z^2 + \frac{64}{3}z + \frac{512}{27}\right) + \left(-8z^2 - \frac{128}{3}z - \frac{512}{9}\right) + \left(9z + 24\right) - 18\n]", "Group like terms:", "- ( z^3 ):\n ( z^3 )", "- ( z^2 ):\n ( 8z^2 - 8z^2 = 0 ) ✅ — the quadratic term vanishes!", "- ( z ):\n ( \frac{64}{3}z - \frac{128}{3}z + 9z = \left(\frac{64 - 128 + 27}{3}\right)z = \frac{-37}{3}z )", "- Constants:\n ( \frac{512}{27} - \frac{512}{9} + 24 - 18 )", "Compute constants step-by-step:", "- ( \frac{512}{27} - \frac{512 \cdot 3}{27} = \frac{512 - 1536}{27} = \frac{-1024}{27} )\n- Convert 24 and −18 to 27ths: ( 24 = \frac{648}{27},\ -18 = -\frac{486}{27} )\n- Sum:\n ( \frac{-1024 + 648 - 486}{27} = \frac{-862}{27} )", "Thus, the transformed function is:", "[\nf(z + \ frac{8}{3}) = z^3 - \frac{37}{3}z - \frac{862}{27}\n]", "### Step 3: Interpret the Simplified Cubic", "The transformed expression:", "[\nf(z) = z^3 - \frac{37}{3}z - \frac{862}{27}\n]", "is a depressed cubic — cubic polynomial without the ( z^2 ) term — and thus denotes a function symmetric about ( z = 0 ) in this shifted coordinate system. This form is optimal for:", "- Finding critical points: Compute derivative ( f'(z) = 3z^2 - \frac{37}{3} ), set to zero:\n [\n 3z^2 = \frac{37}{3} \Rightarrow z^2 = \frac{37}{9} \Rightarrow z = \pm \frac{\sqrt{37}}{3}\n ]\n These are stable and unstable equilibria of the cubic, locate local extrema.", "- Solving ( f(w) = 0 ): Use trigonometric or Cardano’s formula on depressed cubics, as the absence of ( z^2 ) simplifies root expressions.", "- Analyzing concavity and inflection: Second derivative ( f''(z) = 6z ), zero at ( z = 0 ), confirming natural symmetry axis.", "### Conclusion", "By applying the substitution ( w = z + \frac{8}{3} ), we eliminate the quadratic term elegantly, transforming ( f(w) ) into a cleaner cubic in ( z ):", "[\nf(z + \ frac{8}{3}) = z^3 - \frac{37}{3}z - \frac{862}{27}\n]", "This representation reveals the function’s core structure, enabling efficient determination of maxima, minima, and roots—key tools for both theoretical analysis and practical applications. The substitution exemplifies how strategic algebraic transformations unlock deeper understanding in polynomial functions.", "---", "Keywords:\ncubic function analysis, substitution ( w = z + \frac{8}{3} ), eliminate quadratic term, depressed cubic, function transformation, cubic root, calculus of cubics, algebra simplification, ( f(w) = w^3 - 8w^2 + 9w - 18 ), root finding, extremum location, cubic equations, coordinate shift."]
