Polynomial Operations Overview

Overview

This lecture explains how to add, subtract, multiply, and divide polynomial expressions using key algebraic techniques.

Combine like terms: add or subtract coefficients of terms with the same variable and exponent.
To subtract, distribute the negative sign to each term of the second polynomial before combining like terms.
Example: (4x² + 5x + 7) + (3x² - 8x + 12) = 7x² - 3x + 19.

Use the distributive property (FOIL method for binomials) to multiply each term in one polynomial by each term in the other.
For (3x + 5)(2x - 3), FOIL yields 6x² + x - 15 after combining like terms.
When multiplying a binomial (2 terms) by a trinomial (3 terms), expect 2 × 3 = 6 initial terms before combining like terms.
Example: (4x - 2)(x² + 3x - 5) yields 4x³ + 10x² - 26x + 10 after combining like terms.
Multiplying two trinomials gives 9 terms to combine.

Three methods: factoring, long division, synthetic division.
Factoring: Factor numerator and denominator, then cancel common factors.
Long division: Divide highest degree terms, multiply, subtract, repeat until remainder is zero or less degree than divisor.
Synthetic division: Write coefficients, use root of divisor, perform multiplication and addition for new coefficients.

Polynomial — Expression with terms in the form axⁿ, where n is a non-negative integer.
Like Terms — Terms with identical variable parts (same variable and exponent).
Binomial — Polynomial with two terms.
Trinomial — Polynomial with three terms.
FOIL Method — Technique to multiply two binomials: First, Outer, Inner, Last.
Long Division (Polynomials) — Sequence of dividing, multiplying, and subtracting to simplify a polynomial division.
Synthetic Division — Shortcut method for dividing a polynomial by a binomial of the form (x - c).

Practice adding, subtracting, multiplying, and dividing polynomial expressions as shown.
Review problems using all three polynomial division techniques.
Prepare for a quiz on combining like terms and performing FOIL multiplication.