Monomial and polynomials

algebraic expression



Is a combination of symbols used in algebra, containing one or more numbers, variables, and arithmetic operationsExamples of Algebraic Expression
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Monomial

A monomial is an algebraic expression in which the only operations that appear between the variables are the product and the power of the natural exponent.

  • 2x2y3z

    Parts of a Monomial

Coefficient

The coefficient of a monomial is the number that multiplies the variable(s).

Literal Part

The literal part is constituted by the letters and its exponents.

Degree

The degree of a monomial is the sum of all exponents of the letters or variables.

Similar monomials
Two monomials are similar when they have the same literal part.
2x2 y3 z is similar to 5x2 y3 z


Addition of Monomials
Similar monomials can be added.
The sum of the monomials is another monomial that has the same literal part and whose coefficient is the sum of the coefficients.
axn + bxn = (a + b)xn
2x2 y3 z + 3x2 y3 z = 5x2 y3 z
If the monomials are not similar, a polynomial is obtained.
2x2 y3 + 3x2 y3 z


Multiplication of Monomials

The multiplication of monomials is another monomial that takes as its coefficient the product of the coefficients and whose literal part is obtained by multiplying the powers that have the same base.
axn · bxm = (a · b)(xn · x m) = (a · b)xn + m
(5x2 y3 z) · (2 y2 z2) = 10 x2 y5 z3

Division of Monomials

Dividing monomials can only be performed if they have the same literal part and the degree of the dividend has to be greater than or equal to the corresponding divisor.
The division of monomials is another monomial whose coefficient is the quotient of the coefficients and its literal part is obtained by dividing the powers that have the same base.
axn : bxm = (a : b) (xn : x m) = (a : b)xn − m
Division of Monomials
Division of Monomials

Power of a Monomial

To determine the power of a monomial, every element in the monomial is raised to the exponent of the power.
(axn)m = am · xn · m
(2x3)3 = 23 · (x3)3 = 8x9
(−3x2)3 = (−3)3 · (x2)3 = −27x6

Adding polynomial

To add two polynomials, add the coefficients of the terms of the same degree.
P(x) = 7x4 + 4x2 + 7x + 2 Q(x) = 6x3 + 8x +3
Sum of Polynomials
Sum of Polynomials

P(x) + Q(x) = 7x4 + 6x3 + 4x2 + 15x + 5

Multiplication of polynomials


Multiply each monomial from the first polynomial by each of the monomials in the second polynomial.

The polynomials can also be multiplied as follows:
Multiplying Polynomials
Multiplying Polynomials

Square of a Binomial

The square of a binomial is always a trinomial. It will be helpful to memorize these patterns for writing squares of binomials as trinomials.
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2




Exercises
http://www.vitutor.com/alg/polynomials/monomial_worksheets.html