Polynomial

A Polynomial can be expressed in terms that only have positive integer exponents and the operations of addition, subtraction, and multiplication. In other words, it must be possible to write the expression without division. It’s easiest to understand what makes something a polynomial equation by looking at examples and non examples as shown below.

Examples of Polynomial         Explanation:

x² + 2x +5             (Since all of the variables have integer exponents that are positive this is a polynomial)

5x + 1                     (Since all of the variables have integer exponents that are positive this is a polynomial)

5x³ -3x² +6x        (Since all of the variables have integer exponents that are positive this is a polynomial)

6x¼ -3                  (Not a polynomial because a term has a fraction exponent)

(5x +1) ÷ 3x         (Not a polynomial because of division)

5x^-2 +1                  (Not a polynomial, because a term has a negative exponent)

(6x² +3x) ÷ (3x) (Is actually a polynomial because it’s possible to simplify this to 3x + 1 –which of course satisfies the requirements of a polynomial.

Polynomial Equation

Polynomial Equation of Degree n

A polynomial equation is an equation that can be written in the form axⁿ + bx^n-1 + . . . + rx + s = 0,
where a, b, . . . , r and s are constants. The largest exponent of x appearing in a non-zero term of a polynomial is called the degree of that polynomial.

Examples:

1. Consider the equation 3x+1 = 0. The equation has degree 1 as the largest power of x that appears in the equation is 1. Such equations are called linear equations.
2. x² +x-3 = 0 has degree 2 since this is the largest power of x. such degree 2 equations are called quadratic equations or simply quadratics.
3. Degree 3 equations like x³+2x²-4=0 are called cubic.

A polynomial equation of degree n has n roots, but some of them may be multiple roots. For example, consider x³ – 9 x² + 24 x-16 =0.

It is clearly a polynomial of degree 3 and so will have three roots. The equation can be factored as (x-1) (x-4) (x-4) =0. Hence, this implies that the roots of the equation are x=1, x=4, x=4. Hence, the root x=4 is repeated.