Polynomials

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Polynomial is a monomial or sum/difference of monomials.

Monomials are called terms of the polynomial.

Note, that binomial is also polynomial.

Examples of polynomial are:

  • `5+x+x^2`
  • `2x^2+y^3x+zy^3`
  • `10xy^3-zy^2x`

Examples of expressions, that are not polynomials:

  • `x+1/x+3` (second term is not a monomial)
  • `(2-x)/(x+2)` (division is not allowed)

Degree of the polynomial is the largest number among degrees of its monomials.

Leading term is a monomial with the largest degree. Leading coeffcient is a coefficient of the leading term.

For example, in polynomial `4x^4y^3-9z^8y^7+3x^2` first term has degree `4+3=7`, second term has degree `8+7=15` and third term has degree `2`. The largest of numbers 7, 15 and 2 is 15.

Thus, degree of the `4x^4y^3-9z^8y^7+3x^2` is 15. Its leading term is `-9z^8y^7` and leading coefficient is `-9`.

Polynomial in one variable is a polynomial, that contains only one variable.

In general, it can be written as `a_nx^n+a_(n-1)x^(n-1)+...+a_2x^2+a_1x+a_0`, where `n` is positive integer.

Using above definitions, we find, that degree of such polynomial is `n`, leading term is `a_nx^n` and leading coefficient is `a_n`.

Examples of polynomials in one variable:

  • `-2y^5+y^4+3y^2` (degree is 5, leading term is `-2y^5`, leading coefficient is `-2`)
  • `1+x^3+x^2-2x^2` (degree is 3, leading term is `x^3`, leading coefficient is `1`)

Depending on the degree, polynomial in one variable has different names:

  • zero degree: constant. For example, `7`.
  • 1st degree: linear. For example, `2x+3`.
  • 2nd degree: quadratic. For example, `x^2-2x+5`.
  • 3rd degree: cubic. For example, `-4x^3+2x^2-5`.
  • 4th degree: quartic. For example, `3x^4-2x^3+x^2+x+7`.

Exercise 1. Determine whether the following is a polynomial: `x^3y+3y^2-z`?

Answer: yes.

Exercise 2. Determine whether the following is a binomial: `sqrt(y)+x^2y`?

Answer: no.

Exercise 3. Determine whether the following is a binomial: `3x^2+2x+1`?

Answer: yes.

Exercise 4. Find degree of the following polynomial: `x^3+2xy^3+z^5`?

Answer: 5.

Exercise 5. Find degree of the following expression: `x^3+2-x^7+5x^2`?

Answer: 7.