Overview
This lecture covers how to write polynomial functions in standard form and how to factor polynomials using various techniques and special products.
Writing Polynomials in Standard Form
- Standard form requires polynomial terms to be ordered by decreasing exponents of x.
- Rearranging terms so exponents decrease gives the standard form.
- Example: (y = x^4 + 4x^3 - x^2 - 16x - 4) arranges the terms from highest to lowest degree.
- All polynomial functions can be rewritten in standard form regardless of the order of terms.
Special Products in Factoring
- Special products include the square of a binomial, product of sum/difference, and product of binomials or trinomials.
- Factoring rewrites a polynomial as a product of lower-degree polynomials.
- Factoring types: greatest common factor (GCF), difference of squares, sum/difference of cubes, perfect square trinomials, and general trinomials.
Factoring Polynomials – Examples
- Difference of two squares: (4x^2 - 81 = (2x + 9)(2x - 9)).
- Sum of cubes: (8b^3 + 27c^3 = (2b + 3c)(4b^2 - 6bc + 9c^2)).
- Perfect square trinomial: (16x^2 + 72x + 81 = (4x + 9)^2).
- General trinomial: (25m^2 - 20mn + 4n^2 = (5m - 2n)^2).
- Complex factorization: (x^4 - 4x^2 - 45 = (x + 3)(x - 3)(x^2 + 5)).
Combining Like Terms & Multiplication
- When multiplying binomials or expanding expressions, combine like terms to simplify.
- Example: ( (x - 3)(x + 1)^2 ) expands to ( x^3 - x^2 - 5x - 3 ).
- Use distributive property for terms like (2x(x+8) = 2x^2 + 16x).
- Always arrange the final answer in standard form.
Key Terms & Definitions
- Standard Form — A polynomial with terms ordered by decreasing exponents of the variable.
- Factoring — Writing a polynomial as a product of polynomials of lower degree.
- Difference of Squares — A factorable form: (a^2 - b^2 = (a + b)(a - b)).
- Sum/Difference of Cubes — Patterns: (a^3 + b^3 = (a + b)(a^2 - ab + b^2)), (a^3 - b^3 = (a - b)(a^2 + ab + b^2)).
- Perfect Square Trinomial — Takes the form (a^2 + 2ab + b^2 = (a + b)^2).
- General Trinomial — A quadratic of the form (ax^2 + bx + c).
Action Items / Next Steps
- Practice rewriting given polynomials in standard form.
- Factor sample polynomials using the methods demonstrated.
- Review the patterns for special products and apply them to assigned homework problems.