Types of Polynomial Identity
The types of polynomials are based on the degree or the highest power of the identities. Polynomial Identities are classified as follow:
- 2nd Degree Polynomial Identities
- 3rd Degree Polynomial Identities
- n Degree Polynomial Identities
2nd Degree Polynomial Identities
Second-degree polynomial identities consists of the polynomials of degree 2 i.e. they involve the identities where maximum power of a variable in a term is 2.
Following is the list of all 2nd Degree Polynomial Identities:
- (a+b)2=a2+b2+2ab
- (a−b)2=a2+b2−2ab
- (a+b+c)2=a2+b2+c2+2ab+2bc+2ca
- (a−b−c)2=a2−b2−c2−2ab+2bc−2ca
- (x+a)(x+b)=x2+(a+b)x+ab
3rd Degree Polynomial Identities
Third-degree polynomial identities consists of the polynomials of degree 3 i.e. they involve the identities where maximum power of a variable in a term is 3.
Following is the list of all 3rd Degree Polynomial Identities:
- (a+b)3=a3+b3+3ab(a+b)
- (a−b)3=a3−b3−3ab(a−b)
- a3+b3+c3–3abc=(a+b+c)(a2+b2+c2–ab–bc–ca)
n-Degree Polynomial Identities
n degree polynomial identities consists of the polynomials of degree ‘n’ i.e. they involve the identities where maximum power of a variable in a term is ‘n’. Here ‘n’ is any natural number.
Following is the formula used for all n Degree Polynomial Identities:
an-bn = (a-b)[(an−1)+(an−2)b+…+(bn−2)a+(bn−1)]
where n is a natural number
- If n is even (n = 2k)
- If n is odd (n = 2k+1)
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Polynomial Formulas
Following are some more polynomial formulas used in mathematics:
- a2−b2=(a+b)(a−b)
- a3−b3=(a−b)(a2+ab+b2)
- a3+b3=(a+b)(a2−ab+b2)
- a4−b4=(a2−b2)(a2+b2)
- a4−b4=(a+b)(a−b)[(a+b)2−2ab]
- 2(a2+b2)=(a+b)2+(a−b)2
- (a+b)2−(a−b)2=4ab
Polynomial Identities
Polynomial identities are mathematical expressions or equations that are true for all values of the variables involved. These identities are particularly useful in simplifying expressions and solving equations involving polynomials.
These are the equations involving polynomials that hold true for all values of the variables involved. These identities are very useful in simplifying expressions and solving equations more efficiently.
It is an equation that hold for all values of the variables within them. These identities are often used to simplify expressions and solve polynomial equations more easily.
It is an equation that holds for all possible values of the variables involved. It establishes a relation between two or more polynomial expressions, regardless of the specific numerical values of the variables. One common example is the polynomial identity (a+b)2=a2+ 2ab +b2, which remains true for any values of a and b.
Let’s know more about various identities of polynomials, types of polynomial identities, and their proof along with some solved examples for clear understanding.