Polynomial and Its Standard Form
Definition of Polynomial and Its Standard Form
A polynomial is an algebraic expression that is the sum of monomials.
For example: $ 5x^2 y+4xy^2-3; a-2; 4m^3 n+n $
The monomials that make up a polynomial are called the terms. A polynomial (Greek "poly" - many, "nomos" - part) that consists of two terms is called a binomial (Latin "bi" - two), of three - a trinomial (Latin “tri” - three).
The standard form of a polynomial is its expression as a sum of monomials in the standard form, without similar monomials; the terms are arranged from the largest exponent to the lowest.
The degree of a term is the sum of the exponents of its variables.
The degree of a polynomial in the standard form is the largest degree of the terms.
Any polynomial can be expressed in the standard form.
Algorithm for Finding Standard Form of Polynomial
- Find the standard form for each of the monomials that make up the polynomial.
- Combine like terms.
- Arrange the terms from the largest exponent to the lowest.
How to find the standard form for a monomial (see §12 of this Guide).
How to combine like terms (see §2 of this Guide).
The number 0, as well as polynomials identically equal to zero (for example,$0a+0b,z^2-z^2$), are called zero-polynomials. Zero-polynomials are considered to have no standard form, neither degree.
Examples of Problem Solving
Example 1. Simplify the polynomial by writing it in the standard form. Specify the degree of the polynomial:
a) $ x^2 yxy+\frac{1}{2} x^3 y^4-\frac{1}{3} yx^3 y+3xyx^2 y^3 = x^3 y^2+\frac{1}{2} x^3 y^4-\frac{1}{3} x^3 y^2+3x^3 y^4 = $
$ = \left(1-\frac{1}{3}\right) x^3 y^2+\left(\frac{1}{2}+3\right)x^3 y^4 = \frac{2}{3} x^3 y^2+3,5 x^3 y^4 $
Degrees: first term 3+2 = 5, second term 3+4 = 7.
The degree of the polynomial is 7.
b) $ 25abca^2-2a^2 c^3+cac^2 a-12a^3 bc=25a^3 bc-2a^2 c^3+a^2 c^3-12a^3 bc = $
$ = (25-12) a^3 bc+(-2+1) a^2 c^3 = 13a^3 bc-a^2 c^3 $
Both of the terms have the same degree: 3+1+1 = 5 and 2+3 = 5.
The degree of the polynomial is 5.
Example 2. Simplify the polynomial and evaluate it for a = 0,1 and b = 1:
a) $ 6a^{10}+3a-15b-3a^{10}+7a+8b-3a^{10} = (6-3-3) a^{10}+(3+7)a+(-15+8)b = 10a-7b $
Substitute: $10 \cdot 0,1-7 \cdot 1 = 1-7 = -6$
b) $ 8y^3-16+ab-y^3+8ab-7y^3 = (8-1-7) y^3+(1+8)ab-16 = 9ab-16 $
Substitute: $ 9\cdot 0,1\cdot 1-16 = 0,9-16 = -15,1 $
Example 3. Write Χ as a polynomial that has:
a of thousands, b of hundreds, 0 of tens, c of ones
As stated: $ X = 1000 \cdot a+100 \cdot b+10 \cdot 0+1 \cdot c = 1000a+100b+c$
b) a of tens of thousands, b of thousands, c of ones
As stated:
$ X = 10000 \cdot a+1000 \cdot b+100 \cdot 0+10 \cdot 0+1 \cdot c = 10000a+1000b+c $
