Polynomials Types And Factors

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Types of polynomials:

Most commonly we have 3 types of polynomials:

Monomials: The polynomials with only one term. It contains constants and/variables but no + or – signs. For example: 2, x, 2x^3, 5x^2y^3z^2 etc.

Binomials: The polynomials containing exactly two terms are called binomials. The two terms would be connected by a + or – sign. For example: 1 + x, x^2 – y^2, -11x^3 + 8x^2 etc.

Trinomials: When we have exactly three terms in a polynomial we call it a trinomial. For example: x^2 +2x+1, x^2-2xy+y^2, -5x^3+3x^2y^2 – 7z^4 etc.

All polynomials having more than three terms are referred to as ‘polynomials’.

Writing polynomials in one variable:

Conventionally, when we write a polynomial in one variable, the term with the highest power of the variable comes first. This is called a polynomial in descending order or polynomial in standard form. For example, the polynomial, 2x+3+x^2 not a polynomial in descending order. However if we write the same polynomial like this: x^2+2x+3 it becomes a polynomial in descending order. Usually this method of writing polynomials ...
... is traditionally followed.

Factoring polynomials:

Just as with numbers, factoring polynomials means to split the polynomial to factors such that when they are multiplied we get the original polynomial back. Monomials cannot be factored as they are already written as factors. Usually when we factor polynomials we do not factor the constant number portion.
For factoring binomials we usually bring out the common term between the two terms if there is any. For binomials of the type a^2 – b^2 we use the formula (a+b)(a-b) to factor.
For example : 4x^2 + 8x = 4x(x+2), 4x^2 – 9y^2 = (2x+3y)(2x-3y)
Trinomials can be factored the same way by removing the common terms. Also come trinomials are perfect squares and they can be factored like this: For trinomials of the type a^2 + 2ab + b^2, the factors would be (a+b)^2 and for trinomials of the type a^2 – 2ab + b^2 the factors would be (a-b)^2. If the trinomial is not a perfect square, it can be factored using various other methods.

Dividing polynomials:

There are mainly three cases when dividing polynomials:
Case 1: Division of a monomial by a monomial: Quotient of two monomials =(quotient of the numbers) x (quotient of the variables)
Case 2: Division of a polynomial by a monomial: Divide each term of the polynomial by the monomial
Case 3: Division of a polynomial by a polynomial: Use factoring or long division method.

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