Basic Polynomials and and basic calculus 1

for reference only

1. Polynomial → refers to an expression consisting of variables (also called indeterminates), coefficients, and operations of addition, subtraction, multiplication, and non-negative integer exponents. → essentially means "many terms." The word polynomial is derived from two parts: Poly : from the Greek word polys meaning “many” Nomial : from the Latin word "nomen" , meaning "term."

2. Polynomials can be classified into different types based on their characteristics 1. Based on the Number of Terms Monomial: a polynomial with one term ex. 3x 2 , 7, −5y etc . Binomial : two terms ex. x+5, 3x 2 −2x Trinomial : three terms ex. x 2 +2x+1, 4a 3 +3a 2 −6 Multinomial: more than three terms ex. x 4 −3x 3 +2x 2 +x+6

3. Addition: Combine like terms. Ex: 3x 2 + 2x + 5 + x 2 + 4x −7 = 4x 2 + 6x − 2. Subtraction: Subtract coefficients of like terms. Ex: 5x 3 − 3x 2 + 4x − 1 −(2x 3 + x 2 − 5x + 6) = 3x 3 − 4x 2 + 9x − 7 Multiplication: Use the distributive property. Ex: (x + 2) (x 2 − x + 3) = x 3 + x 2 + x + 6. Division: Use polynomial long division. Ex : (x 3 + 2x 2 − 5x − 6) ÷ (x − 2) = x 2 +4x+3 .

4. 2. Based on the Degree (Highest Power of the Variable ) Constant Polynomial : Degree 0. It contains only a constant. Example: 5,−35, -35,−3 Linear Polynomial : Degree 1. Ex: x+2, 3y−7x + 2, Quadratic Polynomial : Degree 2. Ex: x 2 +3x+2,4a 2 −7a+1 Cubic Polynomial : Degree 3. Ex: x 3 −2x 2 +x−5, 4a 2 −7a+1 Quartic Polynomial : Degree 4. Ex: x 4 −x 3 +2x 2 −5x+1 Quintic Polynomial : Degree 5. Ex: x 5 −3x 4 +x 3 −7 Higher-Degree Polynomials : Degree 6 or more. Ex: x 6 +2x 5 −x 4 +4x−3

5.   A linear polynomial (or linear function) can be represented as a straight line on a graph. For example, P(x)=2x+1

6. Quadratic Polynomial / quadratic equation A quadratic equation is generally expressed in the form: ax 2 + bx + c = 0, where a,b , and c are constants and a≠0 Example: x 2 − 3x − 4= 0 Solution : quadratic formula:

7. x 2 − 3x − 4= 0 Vertex = (1.5, -6.25) X=4 X=-1

8. A cubic polynomial is a polynomial of degree 3. It has the general form: f(x) = ax 3 + bx 2 + cx + d Ex. y = x 3 – 6x 2 + 11x – 6; let the (roots) or x= 1, x= 2, x= 3