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POLYNOMIALS - Coggle Diagram
POLYNOMIALS
:star: What is polynomial? An algebraic expression is a collection of variables and real numbers. The most common type of algebraic expression is the polynomial.
The "poly" in "polynomial" means "many". The term "polynomial" refers to sums of many terms, but the term is used to refer to anything from one term to the sum of a zillion terms.
:star: 1)P(x), Q(x), R(x), … are used to show polynomials with variable x.
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:star:2)The highest power in a polynomial is called degree of polynomial and it is shown as d[p(x)] or
deg[p(x)].
:star:3)The coefficient of term with the highest degree of x is called leading coefficients of the polynomial.
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:star:Sum of the coefficients of a polynomial is found by In order to find the sum of the coefficients of a
polynomial we substitute 1 for each variable.
:star:The constant term of a polynomial is found by In order to find the constant term of a polynomial
we substitute zero for each variable.
:star2:Constant Polynomial: For a0∈R and a0≠0 if=a0 then P(x) is said to be a constant polynomial. The degree of a constant polynomial is zero Example:
P(x) =-12
:star2:Zero Polynomial:If P(x)= 0 , it is called zero polynomial.
Degree of zero polynomial is undetermined.
:star2:Let P(x) and Q(x) be two polynomials.If P(x) = Q(x) then P(x) and Q(x) are called equal polynomials
:star2:Adding and Subtracting Polynomials The sum of two polynomials is obtained by adding together the coefficients sharing the same powers
of the variable.The difference of two polynomials is obtained by subtracting together the coefficients sharing the same powers of the variable.
:star2:Multiplication on Polynomials: Let P(x) and Q(x) be two polynomials. By doing multiplication of P(x).Q(x), each terms of P(x) are multiplied with each terms of Q(x) and addition of these algebraic expressions is written in descending or ascending powers of the variable x from the beginning. For example, if a.xm is a term of P(x) and b.xn is a term of Q(x), a.xm. b.xn = a.b.xm+n is a term of P(x).Q(x).
:star: For P(x)≠0 and Q(x)≠0,if deg{(P(x)}=m and deg{(Q(x)}=n then deg{P(x).Q(x)}=m+n
:star2:Factorization of Polynomials Definition: The process of writing a polynomial as a product of two or more polynomials is called factoring.Let P(x), Q(x), and R(x) be polynomials. P(x) and Q(x) are called the factors of R(x) in the equation of R(x) =P(x).Q(x).
:star: Methods of Factorization:
1) Factoring by Using Common Factor
:check:The first step in completely factoring a polynomial is to remove (factor out) any common factors.
:check:For the polynomials A(x), B(x), and C(x) A(x).B(x)+A(x).C(x)=A(x){B(x)+C(x)}
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:check:The area that corresponds to x ² +ax is drawn separately and then together. It means: x² +ax=x(x+a).
:check:2- Factoring by Grouping If there is no common number, common variable or common term in each term of a given polynomial, the terms that have any common factors are made a group by coming together. Then, each group are factor out such that they have the same expression in parentheses. After that, the groups are factor out any common factor.
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:check:5-Factoring by Changing the Variable It is the process of simplifying the polynomial by renaming similar terms with a new variable
:star2: Division on Polynomials:
P(x) is the dividend polynomial.
Q(x) is the divisor polynomial.
B(x) is the quotient polynomial.
K(x) is the remainder polynomial.
:ballot_box_with_check:Division on polynomials is done by following process:
:check:Arrange both the dividend polynomial and the divisor polynomial in descending powers of the variable.
:check: The term with the highest degree of dividend polynomial is divided by the term with the highest degree of divisor polynomial and the result is written as a first term of quotient polynomial.
:check: The first term of quotient polynomial is multiplied with divisor polynomial and the result is subtracted from dividend polynomial.
:check: All these operations are done until the degree of remainder polynomial is smaller than the degree of divisor polynomial.
:star2:Simplifying the Rational Expressions
:Let P(x) and Q(x) be polynomials. The expression in the form of P(x)/Q(x)is called a rational expression.Adding, subtracting, multiplication, and division on rational expressions are made as in rational numbers.When simplifying rational expressions, the numerator and denominator are factor out firstly, then it is simplified if there are any common factors.
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