Polynomials

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Definition of a polynomial:

An algebraic expression of the form a + bx + cx2 + dx3 + ………….. in which a, b, c, d are constants (numbers) and x is a variable, having non-negative integral exponents is called a polynomial in x. The numbers a, b, c, d…….. are also called coefficients (therefore constants as they are numbers that do not change like variable x).

Polynomial

An algebraic expression of the form: a0 + a1 x + a2 x2 + a3 x3 + ………+anxn….. in which a0, a1, a2, a3…an.. are coefficients and x is a variable having non-negative integral exponents is called a polynomial in x.

Degree of a polynomial in one variable:

The greatest power of the variable is the degree of the polynomial.

Examples of polynomials in one variable:

  • 2y + 4 is a polynomial in y of degree 1, as the greatest power of the variable y is 1
  • ax2 +bx + c is a polynomial in x of degree 2, as the greatest power of the variable x is 2
  • 3p4 -10p3 + 2p – 4/3 is a polynomial in p of degree 4, as the greatest power of the variable p is 4
  • 100 is also a polynomial (constant polynomial or monomial – that which contains only one term) in any variable, say x, because 100 is same as 100×0, and we know that x0 = 1.

Degree of a polynomial in two variables:

The degree of a polynomial that contains two or more variables is the greatest of the different sums of the powers of the variables in the various terms of the polynomial. Examples of degree of a polynomial in two or more variables:

In 2x2y+3xy+4, the sum of the powers of variables x and y in: 2x2y is 2 + 1 = 3, 3xy is 1 +1 = 2, 0 in 4.Among the various sums of the powers of the variables in the terms, 3 is the greatest and therefore the degree of the above polynomial is taken as 3.

Caution: Add the powers of the variables in each term to find the greatest sum. Do not add the powers of the variables of all the terms.
What is NOT a polynomial? In the polynomial, the exponent of the variable must be non-negative integers.

Examples of what are NOT polynomials

1/x, x/y, x2/y3, and 2×1/2 + 3x + 5 are not examples of polynomials because: in the first example, the power of x is -1, in the second example, the power of y is -2, in the third example, the power of y is -3.
Each of these powers in the variables is negative, i.e. not positive. In the fourth example, the power of x in 2×1/2 is ½, an exponent which is not an integer. Therefore, the exponents of variables must be both integers and positive to call the algebraic expression a polynomial.
Note: Non-negative integers are positive integers including 0. 0 is neither positive nor negative, so to include it along with the positive integers, it is referred to as non-negative.

Types of Polynomials:

  1. Linear Polynomial: A polynomial of degree 1 is called a linear polynomial. Example : 3x, 5y + 6, 9p + q
  2. Quadratic Polynomial: A polynomial of degree 2 is called a quadratic polynomial. Example: in ax2 + bx + c, the degree is 2
  3. Cubic Polynomial: A polynomial of degree 3 is called a cubic polynomial. Example: a3 +b3 + 3a2b + 3ab2

Standard Form of a Polynomial:

If the terms in a polynomial are written in ascending or descending powers of the variable in it, then the polynomial is said to be in Standard Form.
Examples: 3×3 – 9×2 + 2 is in standard form, as the powers of the variable x are in descending order. -9 + 6x – 4/5 (x3) + x4 is also in standard form, as the powers of the variable x are in ascending order.