\boxed{\dfrac{5}{3}x^6 - 5x^4 + \dfrac{50}{3}x^2 - 18}.

["# Analyzing the Polynomial: (\dfrac{5}{3}x^6 - 5x^4 + \dfrac{50}{3}x^2 - 18)", "The expression (\dfrac{5}{3}x^6 - 5x^4 + \dfrac{50}{3}x^2 - 18) is a sixth-degree polynomial with rational coefficients, exhibiting only even powers of (x). This structure allows for simplification and deeper insight through substitution, making it an excellent candidate for algebraic analysis, graphing, and even optimization. In this SEO-rich article, we break down the components, factorization, and significance of this polynomial.", "---", "## Structure and Simplification\nThe polynomial is:", "[\nf(x) = \dfrac{5}{3}x^6 - 5x^4 + \dfrac{50}{3}x^2 - 18\n]", "Since all terms contain (x^2) except the constant, substitute (u = x^2) to reduce the degree:", "[\nf(x) = \dfrac{5}{3}u^3 - 5u^2 + \dfrac{50}{3}u - 18\n]", "This transformation enables polynomial factoring techniques effective for cubic expressions.", "---", "## Factoring the Substituted Polynomial", "Let’s focus on factoring:", "[\ng(u) = \dfrac{5}{3}u^3 - 5u^2 + \dfrac{50}{3}u - 18\n]", "To simplify, eliminate fractions by multiplying through by 3:", "[\n3g(u) = 5u^3 - 15u^2 + 50u - 54\n]", "Now, apply the Rational Root Theorem to find possible rational roots: factors of 54 over factors of 5, i.e., (\pm1, \pm2, \pm3, \pm6, \pm9, \pm18, \pm27, \pm54, \pm\frac{1}{5}, \dots). Testing (u = 2):", "[\n5(8) - 15(4) + 50(2) - 54 = 40 - 60 + 100 - 54 = 26 <br/>\ne 0\n]", "Testing (u = 3):", "[\n5(27) - 15(9) + 50(3) - 54 = 135 - 135 + 150 - 54 = 96 <br/>\ne 0\n]", "Try (u = \dfrac{3}{5}):", "[\n5\left(\frac{27}{125}\right) - 15\left(\frac{9}{25}\right) + 50\left(\frac{3}{5}\right) - 54 = \frac{135}{125} - \frac{135}{25} + 30 - 54\n= 1.08 - 5.4 + 30 - 54 = -28.32 <br/>\ne 0\n]", "But upon closer inspection, testing (u = 2) was too rough — instead test (u = 3) again carefully:", "Wait — correct computation:", "[\n5(27) = 135,\quad -15(9) = -135,\quad 50(3)=150,\quad -54\n]\n[\n135 - 135 = 0,\quad 0 + 150 = 150,\quad 150 - 54 = 96 <br/>\ne 0\n]", "Try (u = 1):", "[\n5 - 15 + 50 - 54 = -24 <br/>\ne 0\n]", "Try (u = \dfrac{6}{5} = 1.2):", "[\nu^3 = (1.728),\quad u^2 = 1.44\n]\n[\n5(1.728) = 8.64,\quad -5(1.44) = -7.2,\quad \dfrac{50}{3}(1.2) = \dfrac{50}{3} \cdot \dfrac{6}{5} = 20,\quad -18\n]\nBut we multiplied by 3 earlier: evaluate (g(6/5)):", "Since (g(u) = \dfrac{5}{3}u^3 -5u^2 + \dfrac{50}{3}u -18),", "[\ng\left(\frac{6}{5}\right) = \frac{5}{3} \cdot \frac{216}{125} - 5 \cdot \frac{36}{25} + \frac{50}{3} \cdot \frac{6}{5} - 18\n= \frac{1080}{375} - \frac{180}{25} + 20 - 18\n= 2.88 - 7.2 + 20 - 18 = -2.32 <br/>\ne 0\n]", "Instead, return and use synthetic division with candidate root (u = 3) (despite earlier miscalculation). Alternatively, reverse-engineer.", "Let’s factor via grouping after substitution.", "Return to:", "[\ng(u) = 5u^3 - 15u^2 + 50u - 54\n]", "Try factoring by grouping:", "Group as ( (5u^3 - 15u^2) + (50u - 54) = 5u^2(u - 3) + 2(25u - 27) ) — no simplification.", "Try factoring with real root approximation or exact cube root.", "Alternatively, suppose we test (u = 3) again numerically:", "Wait — consider factoring directly via rational root trial. Try (u = 3): already tried, no. Try (u = \frac{9}{5})? Too messy.", "Instead, factor numerically or observe structure.", "But notice: multiply polynomial (g(u)) by 3 to avoid fractions:", "[\n3g(u) = 5u^3 -15u^2 +50u -54\n]", "Now, apply rational root theorem: possible roots: (\pm1, \pm2, \pm3, \pm6, \pm9, \pm18, \pm27, \pm54, \pm\frac{1}{5}, \dots)", "Try (u = 2):\n[\n5(8) -15(4) +50(2) -54 = 40 - 60 + 100 - 54 = 26 <br/>\ne 0\n]", "Try (u = \dfrac{3}{1} = 3): too big\nTry (u = \dfrac{9}{5} = 1.8):", "[\nu^2 = 3.24,\quad u^3 = 5.832\n]\n[\n5(5.832) = 29.16,\quad -15(3.24) = -48.6,\quad 50(1.8) = 90,\quad -54\n]\n[\n29.16 - 48.6 = -19.44,\quad -19.44 + 90 = 70.56,\quad 70.56 - 54 = 16.56 <br/>\ne 0\n]", "Try (u = 1.5):", "[\nu^3 = 3.375,\quad u^2 = 2.25\n]\n[\n5(3.375) = 16.875,\quad -15(2.25) = -33.75,\quad \frac{50}{3}(1.5) = 25,\quad -18\n]\nBut in scaled:\n[\n5(3.375) = 16.875,\quad -15(2.25) = -33.75,\quad \frac{50}{3}(1.5) = 25,\quad -18\n]\nSum: (16.875 - 33.75 = -16.875,\quad -16.875 + 25 = 8.125,\quad 8.125 - 18 = -9.875 <br/>\ne 0\n]", "Try (u = 3) again inaccurately — actually, use calculator fallback.", "Alternatively, use substitution insight.", "Notice all powers are multiples of (u) except constant. Try factoring as cubic in (u).", "But observe: coefficients hint a perfect cube or symmetric form.", "Assume (g(u)) factors as product of linear & quadratics.", "Use derivative to analyze nature:", "[\ng'(u) = 15u^2 -30u +50\n]", "Discriminant: (900 - 4(15)(50) = 900 - 3000 = -2100 < 0), so (g'(u) > 0), (g(u)) strictly increasing.", "Thus, exactly one real root, and two complex conjugate roots.", "Use numerical approximation:", "Try (u = 2): (g(2) = 5(8) -15(4) +50(2) -54 = 40 - 60 + 100 - 54 = 26)", "Try (u = 1): (5 -15 +50 -54 = -24)", "So root between 1 and 2.", "Try (u = 1.2):\n(u^2 = 1.44), (u^3 = 1.728)\n[\ng(1.2) = 5(1.728) = 8.64,\quad -15(1.44) = -21.6,\quad \frac{50}{3}(1.2) = 20,\quad -18\n]\n[\n8.64 -21.6 = -12.96,\quad -12.96 + 20 = 7.04,\quad 7.04 - 18 = -10.96\n]", "Try (u = 1.8):\n(u^3 = 5.832), (u^2 = 3.24)\n[\n5(5.832) = 29.16,\quad -15(3.24) = -48.6,\quad \frac{50}{3}(1.8) = 30,\quad -18\n]\n[\n29.16 -48.6 = -19.44,\quad -19.44 + 30 = 10.56,\quad 10.56 - 18 = -7.44\n]", "Still negative — wait, earlier at (u=2) was +26, so root between 1.8 and 2.", "At (u=1.9):\n(u^3 = 6.859), (u^2 = 3.61)\n[\n5(6.859) = 34.295,\quad -15(3.61) = -54.15,\quad \frac{50}{3}(1.9) \approx 31.67,\quad -18\n]\n[\n34.295 -54.15 = -19.855,\quad -19.855 + 31.67 = 11.815,\quad 11.815 - 18 = -6.185\n]", "Still negative.", "At (u=1.95):\n(u^2 = 3.8025), (u^3 \approx 7.414)\n[\n5(7.414) = 37.07,\quad -15(3.8025) = -57.0375,\quad \frac{50}{3}(1."]