Factoring Polynomials Basics

Polynomial Basics

• A polynomial consists of terms or a singular term with whole number exponents on variables.
• Examples of polynomials include x^2 + 3x and 4y^3 - 2y + 1.

Key Definitions

• Factor: An integer or polynomial that divides another evenly. In algebra, factoring involves expressing a polynomial as a product of its prime factors.
• Factoring is crucial for simplifying polynomial expressions and solving equations.

Factoring by Grouping

• Recognize patterns within polynomial expressions to group terms effectively.
• Use the method of factoring by grouping to simplify polynomials into products of binomials.

Example Factorizations

• Factorization of jm + jn + km + kn results in (j + k)(m + n).
• The expression ab + 2a + 3b + 6 can be factored to (a + 3)(b + 2).
• The polynomial x^2 + xy - 2x - 2y simplifies to (x + y)(x - 2).

More Complex Factorizations

• The expression mn - 4m - 5n + 20 factors as (m - 5)(n - 4).
• For ab + 4 + a + 4b, the factored form is (a + 4)(b + 1).
• The polynomial y^2 + 10z - 10y - yz can be factored to (y - z)(y - 10).

Higher Degree Polynomials

• The expression n^3 - n^2 + 3n - 3 simplifies to (n^2 + 3)(n - 1).
• For mr + ns - nr - ms, the factorization results in (m - n)(r - s).
• The polynomial z^3 - 2z^2 + 9z - 18 factors to (z^2 + 9)(z - 2).

Important Notes

• Factors should be listed alphabetically when placing into grids or completing factorization tasks.
• Familiarity with common factorization patterns speeds up the simplification of more complex expressions.