Factoring Polynomials refers to factoring a polynomial into irreducible polynomials over a
given field. It gives out the factors that together form a polynomial function. A polynomial
function is of the form xn + xn -1 + xn - 2 + . . . . + k = 0, where k is a constant and n is a
Polynomials are expressions that are formed by adding or subtracting several variables called
monomials. Monomials are variables that are formed with a constant and a variable of some
degree. Examples of monomials are 5x3, 6a2. Monomials having different exponents such as
5x3 and 3x4 cannot be added or subtracted but can be multiplied or divided by them. Any
polynomial of the form F(a) can also be written as
F(a) = Q(a) x D (a) + R (a)
using Dividend = Quotient x Divisor + Remainder. If the polynomial F(a) is divisible by Q(a),
then the remainder is zero. Thus, F(a) = Q(a) x D(a). That is, the polynomial F(a) is a product
of two other polynomials Q(a) and D(a). For example, 2t + 6t2 = 2t x (1 + 3t).
Variables, Exponents, Parenthesis and Operations (+, -, x, /) play an important role in
factoring a polynomial.
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Factorization by dividing the expression by the GCD of the terms of the given expression:
GCD of a polynomial is the largest monomial, which is a factor of each term of the polynomial.
It involves finding the Greatest Common Divisor (GCD) or Highest Common Factor (HCF) of
the terms of the expression and then dividing each term by its GCD. Therefore the factors of
the given expression are the GCD and the quotient thus obtained.
What is Factoring :- Recall the distributive law (BR-9) from the Basic Rules of Algebra
section. This is generally called "expanding" while doing this rule in reverse is called
"factoring". The expressions to either expand or factor are usually more complex than a(b + c)
= ab + ac, but the same procedure is followed (recall the "FOIL" method discussed in the
Basic Rules of Algebra section).
Essentially, factoring is rewriting an expression as a product of 2 or more expressions. [Recall
back to elementary mathematics where factoring a number, such as 15, would be writing it as
the multiple of two other numbers, such as 15 = (5)(3) where both 5 and 3 are factors of 15.]
While a polynomial is an expression involving powers of x (or any variable) that involves only
addititon and multiplication as the types of arithmetic operations used. Each term in a
polynomial can be written as axj where a is a real number and j is a non-negative integer.
There is more information about this in the Polynomials and Roots section.
Common Factors :- One of the most basic ways to factor an expression is to "take out a
common factor". If every term in an expression has several factors, and if every term has at
least one factor that is the same, then that factor is called a common factor. If this is the case,
then the common factor can be "taken out" of every term and multiplied by the whole
Perfect Squares :- Any expression of the form: x2 + 2ax + a2 is a perfect square because x2
+ 2ax + a2 = (x + a)2 (i.e. it can be writen as (something)2). [You can check to see that this is
correct by expanding (x + a)2]. To recognize an expression as a perfect square or not, you
should first see if the constant term is a square number (i.e. can you take the square root of it
and get an integer for an answer?).
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Transitive Property of Equality
Transitive Property of Equality
Definition: Property is the reasonable laws for real numbers in mathematics. Here we practice
the equality property briefly. Thus the equality properties are used to activate, stable the
equations. Generally, equality is referred as follows,
--- m = n indicates m is equal to n.
--- m ≠ n indicates m does not equal n.
Thus, the properties of equalities contain the following properties.
1. Balance equation relation property
a) Addition property b) Subtraction property
c) Multiplication property d) Division property
2. Equivalence relation property
a) Reflexive property b) Symmetric property c) Transitive property
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3. Distributive property
Here the transitive property is explained briefly as follows,
Generally, the transitive property deals with following categories,
1) Transitive property of equalities
2) Transitive property of inequalities
1) Transitive property of equalities :
If m, n, and o are real numbers, then transitive property of equalities states that m = n and n =
o then m =o. Thus, the two quantities equal to the same quantity are identical to each other.
Otherwise, transitive property of equalities referred as, if two numerical are equal to same
number, then all numerical are equal to each other.
2) Transitive property of inequalities :
The transitive property of inequalities states as follow,
--- If x < y and y < z, then x < z.
--- If x ≤ y and y ≤ z, then x ≤ z.
--- If x > y and y > z, then x > z.
--- If x ≥ y and y ≥ z, then x ≥ z
Also the transitive property of inequalities said that if a number is less than or equal to a
second number, and the second number is less than or equal to a third number, then the first
number is also less than or equal to the third number.
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