Definition of a Polynomial
What is a polynomial?
- A polynomial is a sum of terms with non-negative integer powers of x
- the highest power of x is its degree
- The following are polynomials:
- (degree 3)
- (degree 6)
- (degree 0)
- The following are not polynomials:
- (the can be written which has a negative power)
- (the can be written which has a non-integer power)
- ( is not a power of x)
- There are words to name different types of polynomials:
-
degree name 0 constant 1 linear 2 quadratic 3 cubic 4 quartic 5 quintic ... ...
-
- For the polynomial
- ... the degree is 4 (it's a quartic)
- ... the coefficient of x2 is -3
- ... the quadratic term (or term involving x2) is -3x2
- ... the coefficient of x3 is 0
- ... the constant is 8
How do I add / subtract / multiply polynomials?
- Adding and subtracting two polynomials requires collecting like terms
- for example, gives
- whereas gives
- the second bracket expands to
- Multiplying two polynomials can be done using a grid method (multiplying rows by columns then combining terms to get the answer)
- for example, can be done as follows:
-
x3 4x x2 x5 4x3 8x 8x4 32x2 3 3x3 12x - add any like terms that arise:
- the final answer is therefore