Factor using a quadratic pattern

Learning Objectives Identify polynomials and other expressions that exhibit a quadratic pattern. Use algebraic substitution (u-substitution) to transform a higher-degree polynomial into a standard quadratic form. Factor polynomials of the form ax^(2n) + bx^n + c completely. Factor expressions with rational exponents or trigonometric functions that follow a quadratic pattern. Solve polynomial equations by first factoring them using a quadratic pattern. Apply secondary factoring techniques, such as the difference of squares, after an initial quadratic pattern factorization. Ever looked at a beastly polynomial like x⁶ + 2x³ - 15 and thought it was impossible to factor? What if it's just a simple quadratic in disguise? 🤔 This tutorial will teach you a powerful pattern rec...

TermDefinitionExample Quadratic Pattern (or Quadratic Form)An expression that can be written in the form au² + bu + c, where 'u' is itself an algebraic expression (like x², x³, sin(x), or eˣ). The key feature is that the exponent of the leading term's variable part is exactly double the exponent of the middle term's variable part.The polynomial x⁴ - 5x² + 6 is in a quadratic pattern because if we let u = x², it becomes u² - 5u + 6. U-SubstitutionAn algebraic technique where a part of an expression is replaced with a single variable, typically 'u', to simplify the expression's structure and reveal an underlying pattern.In x⁶ + 7x³ + 12, we let u = x³ to transform it into the simpler quadratic u² + 7u + 12. Back-SubstitutionThe final step in the u-substitu...

The Quadratic Pattern Structure ax^{2n} + bx^n + c This is the general form of an expression that follows a quadratic pattern. To factor it, identify the middle term's variable part, xⁿ, and use it for substitution. The Substitution Rule Let u = x^n. Then ax^{2n} + bx^n + c \rightarrow au^2 + bu + c This is the core transformation. By setting 'u' equal to the variable part of the middle term, the entire expression simplifies into a standard quadratic trinomial that you already know how to factor. Difference of Squares (Recursive Application) a^2 - b^2 = (a - b)(a + b) This rule is often required for complete factorization. After back-substituting, you may find that one or more of your new factors is a difference of squares and can be factored further....