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Polynomials

Polynomial Long Division

Polynomials are math equations that are multinomial. They have two or more terms with each term separated by a plus (+) or minus (-) sign, and do not have any letters under a radical (√ ) or in a denominator. Their exponents are integer and  are always positive:

3x + 5ab + 2y + 4 is a four term polynomial.

6x + 5ab + 4 is a three term polynomial.

√x - 3 is not a polynomial, “x” is under a radical.

3/x + 6 is not a polynomial, “x” is a denominator.

Degree of Polynomial

The degree of polynomial that has a single literal factor is the term having the highest degree:

4 x2 - 2x + 4 is a second degree polynomial determined by the “ x2 ” literal factor.

If there are two or more variables in a term the degree of polynomial is the sum of the exponents of the variables:

2ax3 - 4x + 6 is a fourth degree polynomial determined by the sum of the “a” and “ x3 ” factors.

3x + 5ab + 2b + 4 is a second degree polynomial, “a” and “b” each count as one from the “5ab” term.

Polynomial Function

Let “R” be the set of real numbers. The zero polynomial function is defined to be function “g” as:

g(x) = 0 ; For all “x” in “R”, with a degree not defined.

If some number “R” is a solution of g(x) = 0, then “R” is called a zero of the polynomial. If “R” is a solution of g(x) = 0, then g(R) = 0. By substituting “R” in for “x” results in the value 0 if “R” is a zero of g(x).

Let “R” be the set of real numbers.

For all polynomial functions excluding the zero function:

f(x) = an xn + an -1 xn -1 + …+ a1 x + a0

For all “x” in “R”, and “x” ≠ 0, and “n” is a nonnegative integer. Each “a” is a coefficient of the polynomial function:

a0 is a constant term.

an is the leading coefficient.

fn = 0, 1, 2, respectively we obtain:

f(x) = cx0 = c, the constant function, is not a polynomial, n = 0

f(x) = ax + b, the linear equation, n = 1

f(x) = ax2 + bx + c, the quadratic equation, n = 2

n = 3 is a cubic function

n = 4 is a quartic function

All polynomials are continuous functions. When a polynomial is drawn or is displayed as a graph the resulting line does not break, skip or gap. If you were using pencil and paper the line could be drawn from its beginning to end without lifting the pencil from the paper.

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