Polynomials

Polynomials

Polynomial comes from poly – means “many” and -nomial (means “term”). So, it is many terms.

Polynomials are algebraic expressions that have real numbers and variables. The variables include addition, subtraction and multiplication. Division and square roots cannot be involved in variables.

An expression of form p(x) = a0+a1x+a2x2…..+anxn, where an ≠ 0, is called polynomial in x of degree n.

So, polynomials can include:

  • Constants (e.g. 2, 5, -10, ½ ….)
  • Variables (e.g. x and y)
  • Exponents (e.g. 2 in x2), only 0, 1, 2, 3…. are allowed, not negative.

Let us consider a few examples and check whether they are polynomials are or not?

  1. 5xy-2 – No. Because -2 exponent is not allowed. Only 0, 1, 2, 3….. are allowed.
  2. x/2 – Yes. Because you can divide by a constant.
  3. 3/(x+2) – No. Because division by a variable is not allowed.
  4. 1/x – No. Because division by a variable is not allowed.
  5. 10x/2 – Yes. Because you can divide by a constant.
  6. √x – No. Because exponent is ½. Only 0, 1, 2, 3….. are allowed.
  7. √2 – Yes. Because it is constant.

Polynomials have special names like monomial, binomial, trinomial which have 1, 2 and 3 terms respectively.

5b3c, 2a, a2 are monomials.

2a+7b, 3x+5, 9y - 2y2 are binomials.

-3x2+2+3x, 9y - 2y2-y are trinomials.

A polynomial can have no variable, one variable or two or more variables.

The degree of the polynomial is the highest exponent of the variable. Polynomials with degree 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.

Example of polynomial in an equation

x2 - 5x - 4 
 

Term

Numerical Coefficient

x2
-5x
-4

 1
 -5
 -4

 

The standard form of writing a polynomial is writing in decreasing order of terms. The largest term or the term with highest exponent is written first and is called the leading term in the polynomial.

For example, x5 + 2x3 + 4x2 – 10.

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