Overview
This lecture covers how to add, subtract, multiply, and divide polynomial expressions using methods such as combining like terms, distribution, FOIL, factoring, long division, and synthetic division.
Adding and Subtracting Polynomials
- Combine like terms (terms with the same variable and exponent) when adding or subtracting polynomials.
- Distribute negative signs to all terms when subtracting one polynomial from another before combining like terms.
- If coefficients are outside the parentheses, distribute them before combining like terms.
Multiplying Polynomials
- Use the FOIL (First, Outer, Inner, Last) method to multiply two binomials (expressions with two terms).
- Squaring a binomial (e.g., (2x - 5)^2) means multiplying the binomial by itself and then combining like terms.
- When multiplying a binomial by a trinomial, multiply every term in the binomial by each term in the trinomial, resulting in six terms before combining.
- Multiplying two trinomials (each with three terms) will initially produce nine terms; then combine like terms.
Dividing Polynomials
- To divide a trinomial by a binomial, you can factor the numerator and cancel common factors with the denominator.
- Polynomial long division involves dividing the terms, multiplying, subtracting, and bringing down terms until the remainder is zero or degree is less than divisor.
- Synthetic division is a shortcut for division when dividing by a binomial of the form (x - k); use coefficients and arithmetic steps to find the result.
Key Terms & Definitions
- Polynomial — an expression consisting of variables, coefficients, and non-negative integer exponents.
- Binomial — a polynomial with exactly two terms.
- Trinomial — a polynomial with exactly three terms.
- Like Terms — terms with the same variable(s) raised to the same power(s).
- FOIL Method — multiplies two binomials by multiplying First, Outer, Inner, and Last terms.
- Factoring — rewriting a polynomial as a product of simpler polynomials.
- Long Division (Polynomials) — divides the polynomial stepwise, similar to arithmetic long division.
- Synthetic Division — a shortcut method for dividing polynomials by x - k.
Action Items / Next Steps
- Practice combining like terms in addition and subtraction of polynomials.
- Work additional problems using FOIL, distribution, and multiplying binomials/trinomials.
- Try dividing polynomials using factoring, long division, and synthetic division.