Polynomials are named after the highest power of the variable; A polynomial in standard form is represented as: [tex]ax^n + bx^{n-1} + cx^{n-2} +......+d[/tex].
An example of a third degree polynomial is [tex]2x^3 + 4x^2 - 3x + 1[/tex]
Polynomials are closed under:
- Addition
- Subtraction
- Multiplication
Polynomials are not closed under division.
The conditions that make a third degree polynomial are:
- The highest coefficient of the variable must be 3
- There must be at least one term in the polynomial
So, a third degree polynomial in standard form is represented as:
[tex]ax^3 + bx^2 + cx + d[/tex]
Where
[tex]a \ne 0[/tex]
An example of a third degree polynomial is:
[tex]2x^3 + 4x^2 - 3x + 1[/tex]
It is in standard form because:
- The highest degree of x is 3
- The polynomial has a decreasing power of x from 3 to 0
Polynomials are closed under addition, subtraction and multiplication, because the operations will give rise to another polynomial.
Take for instance:
[tex](2x^2 + 3x + 4) + (3x^2 - 2x + 1) = 5x^2 + x + 5[/tex] -- addition
[tex](4x^2 + 7x + 4) - (3x^2 - 2x + 1) = x^2 + 5x + 5[/tex] -- subtraction
[tex](x + 1) \times (x^2 + 4x + 3) = x^3 + 5x^2 + 7x + 3[/tex] -- products
Notice that the above operations gives another polynomial. Hence, we can conclude that polynomials are closed under the three operations
However, polynomials are not closed under division because the result of the division operation can equal to a non polynomial.
Take for instance:
[tex]\frac{8x^4}{2x^8} = 4x^{-4}[/tex]
Notice that [tex]4x^{-4}[/tex] is not a polynomial.
Hence, polynomials are not closed under division.
Read more about polynomials at:
https://brainly.com/question/11391029
