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POLYNOMIALS, image, image, image - Coggle Diagram
POLYNOMIALS
Definition
An expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
Terms
Coefficients
If the variable in a term is multiplied by a number, then this number is called the coefficient.
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Degree
The highest power in a polynomial is called degree of polynomial, the power on the variable x. It is shown as d[p(x)] or deg[p(x)].
Constant Term
The variable having a power of zero, it will always evaluate to 1, so it's ignored because it doesn't change anything. It's value never changes, so it is called the "constant" term.
Polynomials
Zero Polynomial
A polynomial having value zero (0) is called zero polynomial. Degree of zero polynomial is undetermined.
Equal Polynomial
Two polynomials are equal if and only if they have the same degree and corresponding terms have equal coefficients.
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Algebraic Operations
Division
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. The first term of quotient polynomial is multiplied with divisor polynomial and the result is subtracted from dividend polynomial.
All these operations are done until the degree of remainder polynomial is smaller than the degree of divisor polynomial.
When a polynomial P(x) is divided by (x - a), the remainder is P(a).
Multiplication
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.
Adding and Subtracting
The difference of two polynomials is obtained by subtracting together the coefficients sharing the same powers of the variable.
The sum of two polynomials is obtained by adding together the coefficients sharing the same powers of the variable.
Factorization
Three Terms Polynomial
For a≠0 and a, b, c ∈ R, by factoring three term polynomials of the form ax² + bx + c, we look at the factors of a and c.
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Using Common Factor
Determine the greatest common factor of the given terms. The greatest common factor or GCF is the largest factor that all terms have in common.
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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.
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