What are Polynomials and Monomials in Algebra?

Question

What are Polynomials and Monomials, their basics, problems, solutions, etc.

Answer 1

Monomial: a product of a nonzero constant and powers of variables with a nonnegative integer exponent

Polynomial: a sum of multiple monomials, including all monomials

Coefficient: the nonzero constant multiplied to a term; includes integer, rational number, real number, and complex number

Degree of a monomial: the sum of all the exponents of the variables

Degree of a polynomial: the degree of a term with the highest degree in that polynomial

Univariate: a single variable

Multivariate: multiple variables

When A, B, C are polynomial:

1. A + B = B + A, AB = BA
2. (A + B) + C = A + (B + C), (AB)C = A(BC)
3. A(B + C) = AB + AC, (A + B)C = AC + BC

A polynomial can be expressed using other polynomials with lesser degrees: A = BQ + R

1. A: the original polynomial
2. B: a nonzero polynomial
3. Q: the quotient
4. R: the remainder

In this case, R’s degree is lower than B’s degree.

Basic product formulas:

1. (a ± b)² = a² ± 2ab + b²
2. (a + b)(a - b)=a² - b²
3. (ax + b)(cx + d) = acx² + (ad + bc)x + bd
4. (x + a)(x + b)(x + c) = x³ + (a + b + c)x² + (ab + bc + ca)x + abc
5. (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
6. (a ± b)³ = a³ ± 3a²b + 3ab² ± b³
7. (a ± b)(a² ± ab + b² ) = a³ ± b³
8. (a + b + c)(a² + b² + c² - ab - bc - ca) = a³ + b³ + c³ - 3abc
9. (a² + ab + b²)(a² - ab + b²)= a⁴ + a²b² + b⁴

Expansion of the formulas:

1. a² + b² = (a ± b)² ± 2ab
2. a³ + b³ = (a+b)³ - 3ab(a + b) = (a + b)(a² - ab + b²)
3. ab + c(a + b) = (a + c)(b + c) - c²
4. xy + ax + by = (x + b)(y + a) - ab

Greatest Common Divisor: the factor with the highest possible degree of the given polynomials

Least Common Multiple: the polynomial with the lowest possible degree that includes the given polynomials as its factors

Identity: equality relating one mathematical expression to another mathematical expression such that it produces the same value for all values of the variables within a certain range of validity.

Basic identity applications:

1. ax + b = 0 for any value of x ⇔ a = 0,b = 0
2. ax + b = cx + d for any value of x ⇔ a = 0, b = 0

f(x) = (ax + b)Q(x) + R

The above equality is an identity

f(-b/a) = {a(-b/a) + b}Q(x) + R = R

f(-b/a) = 0 ⇔ f(x) = (ax + b)Q(x)

How to find factors of a univariate polynomial with integer powers of a variable:

f(x) = axⁿ+... +b

Look for all the f values of x = ± (þ/α), where α is a divisor of a and β is a divisor of b.

A = B:

1. A + M = B + M
2. AM = BM

ax = b:

1. a ≠ 0: x = a/b
2. a = 0, b ≠ 0: no solution
3. a = 0, b = 0: infinite solutions

When dividing by a term, check to see if the term you are dividing by is 0.

(A, B are real numbers.)

AB = 0 ⇔ A = 0 or B = 0

A² + B² = 0 ⇔ A = 0, B = 0

Answer 2

monomial is one term with a complex coefficient, variable base and a positive integer or zero power: like 3x^2 or ix^3

polynomial is more than one monomial. add them together such as a binomial: 2x^2 + 4x

graph them and their x intercepts are the real zeros, solutions or roots

if no x intercept the zeros are imaginary

use a graphing calculator or solve with algebra. but algebra gets difficult or nearly impossible with degree higher than 2. degree = the power of the highest powered term. so graphing is a good way to solve them with a calculator