Solution: Let $ u = \sqrt{x} $, so $ x = u^2 $, and $ P(x) = u^2 \cdot u - 5u^2 + 6u = u^3 - 5u^2 + 6u $.
["SEO Meta Description:\nTransform complex polynomial expressions with the smart substitution $ u = \sqrt{x} $, simplifying $ P(x) = u^3 - 5u^2 + 6u $ to make solving polynomial equations easier and more intuitive for math students and educators.", "---", "Transforming Polynomial Equations: Let $ u = \sqrt{x} $ to Simplify $ P(x) = u^3 - 5u^2 + 6u $", "When tackling polynomial equations involving square roots, direct substitution often reveals elegant solutions. One effective method is letting $ u = \sqrt{x} $, transforming the original expression into a clean cubic polynomial:", "$$\nP(x) = u^3 - 5u^2 + 6u\n$$", "By substituting $ x = u^2 $ and redefining the polynomial in terms of $ u $, we shift from a radical-based expression to a standard cubic polynomial:", "$$\nP(u) = u^3 - 5u^2 + 6u\n$$", "### Why This Substitution Works", "This substitution streamlines the analysis by converting ambiguous radical expressions into familiar algebraic forms. The original function $ P(x) $, which may initially appear complex due to $ \sqrt{x} $, becomes more manageable when expressed in $ u $, especially when factoring or solving for roots.", "### Factoring the Polynomial", "Begin by factoring $ P(u) = u^3 - 5u^2 + 6u $:", "$$\nP(u) = u(u^2 - 5u + 6)\n$$", "Now factor the quadratic:", "$$\nu^2 - 5u + 6 = (u - 2)(u - 3)\n$$", "So the full factored form is:", "$$\nP(u) = u(u - 2)(u - 3)\n$$", "### Finding the Roots", "Setting $ P(u) = 0 $ gives:", "$$\nu(u - 2)(u - 3) = 0\n$$", "Thus, the solutions are:", "- $ u = 0 $\n- $ u = 2 $\n- $ u = 3 $", "### Reverting Back to $ x $", "Recall $ x = u^2 $. Substitute each value of $ u $ to find $ x $:", "- For $ u = 0 $: $ x = 0^2 = 0 $\n- For $ u = 2 $: $ x = 2^2 = 4 $\n- For $ u = 3 $: $ x = 3^2 = 9 $", "### Practical Applications", "This substitution technique is especially useful in calculus, optimization, and solving real-world modeled problems involving square roots. By transforming the polynomial into a simpler cubic form, students and professionals can apply root-finding techniques, graphing methods, or numerical approaches with greater confidence.", "### Summary", "Let $ u = \sqrt{x} $ to convert $ P(x) = (\sqrt{x})^3 - 5(\sqrt{x})^2 + 6\sqrt{x} $ into:", "$$\nP(u) = u^3 - 5u^2 + 6u\n$$", "Factoring yields $ P(u) = u(u - 2)(u - 3) $, so the roots are $ u = 0, 2, 3 $, leading to $ x = 0, 4, 9 $. This elegant substitution clarifies complex radical-containing polynomials, making them easier to analyze and solve.", "---", "Keywords:\npolynomial substitution, let $ u = \sqrt{x} $, simplify $ u^3 - 5u^2 + 6u $, solve radical equations, polynomial factoring, algebraic substitution, mathematical methods, calculus prep", "Target Audience:\nHigh school and college math students, educators, and self-learners studying algebra, polynomial equations, or calculus.", "SEO Tips:\nTarget long-tail keywords like “solve radical polynomial by substitution” and use internal linking with related terms such as polynomial simplification and cubic equations. Optimize headings, bullet points, and meta description with primary keywords for strong search visibility."]
