Polynomial Expressions

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What is a polynomial? The prefix poly suggests a multiplicity of something. In fact, a polynomial is an expression with a number of terms, in which

  • only the operations of addition, subtraction and multiplication have been used. Division by constants is allowed.

  • the exponents of the variables are non-negative integers.

A polynomial is made up of terms, and each term has a coefficient. For example, in the polynomial \({x^2} + 2x + 1,\)there are three terms, and the respective coefficients are\(1,\,\,2,\,\,1\). The constant term can be thought of as the coefficient of x raised to the power 0. For example, \(1 = 1{x^0}\).

We note that polynomials can have any number of terms (even one or two), and they may not even contain any variables. For example, the expressions 0 or x (both containing just one term) are also polynomials. In fact, the expression 0 is a constant polynomial, and it has a special name: the zero polynomial.

We will consider a number of examples of expressions, and in each case decide which is a polynomial and which is not.

  1. The expression \({x^2} + 3x + 2\)is a polynomial, as it satisfies both the constraints we mentioned above.

  2. The expression \({x^2} + 3\sqrt x + 1\) is not a polynomial, as we have a non-integer exponent of the variable in the second term. Note carefully that the constraint of integer exponents applies only to exponents of the variable(s), and not of any constants. For example, the expression \({x^2} + \sqrt 3 x + 1\) is a perfectly valid polynomial!

  3. The expression \({x^3} + {y^3} + {z^3} + 3xyz\) is a polynomial with three variables.

  4. The expression \(y + \sqrt {{x^2} + {y^2}} \) is not a polynomial as the constraint of integer exponents of the variables is not satisfied.

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