Algebra Know-It-ALL

438 Polynomial Equations in Real Numbers

Polynomial Standard Form

A higher-degree equation in polynomial standard form contains a sum of multiples of the

variable raised to powers in descending order on the left side of the equals sign. The right side

of the equals sign has 0 all by itself.

General polynomial equation

The polynomial standard form of a higher-degree equation can be written as

anxn+an-1xn-1+an-2xn-2+ ··· +a 1 x+b= 0

where a 1 ,a 2 ,a 3 , ... an are coefficients, b is the stand-alone constant, and n is a positive integer

greater than 3. Here are some examples:

6 x^4 − 3 x^3 + 3 x^2 + 2 x+ 5 = 0

3 x^5 − 4 x^3 = 0

− 7 x^7 − 5 x^4 + 3 x^3 −x^2 − 29 = 0

− 4 x^11 = 0

In all but the first of these equations, some of the coefficients are equal to 0. The coefficient

an, by which xn is multiplied, can never be 0 in an nth-degree polynomial equation. If you set

an= 0, you end up with

0 xn+an-1xn-1+an-2xn-2+ ··· +a 1 x+b= 0

That’s the polynomial standard form for a single-variable equation of degree n− 1:

an-1xn-1+an-2xn-2+ ··· +a 1 x+b= 0

Mutants in the nth degree

As you can imagine, many single-variable equations can be morphed into the polynomial

standard form. Here are some examples:

x^2 = 2 x+ 3 x^7 − 4 x^15

x= 4 x^21 − 7 x^17 + 2 x^11 + 2 x^7

x^8 + 4 x^6 + 7 x^4 −x^2 + 3 =x+x^3 +x^5 +x^7

The only requirements for membership in the “single-variable nth-degree equation club” are

that the equation be convertible into polynomial standard form, and that n be a positive inte-

ger. If n= 1, we have a first-degree equation; if n= 2, we have a quadratic; if n= 3, we have a

cubic; if n > 3, we have a higher-degree equation.