What are Polynomials?

A polynomial in the variable x is an algebraic expression consisting of one or more terms, where each term is of the form axn.

Here: 1. a is a real number (coefficient) where a ≠=0 .

2. n is a wholeprime number (non-negative integer: 0, 1, 2, 3, ...).

A polynomial can have multiple terms added or subtracted, but it must not contain:

Why is 1y−1 not a polynomial?

• The denominator contains a variable (y-1) , which introduces division by a variable.

• A polynomial cannotcan have variables in the denominator.

Now, look at the polynomial px = 3x7−4x6+x+9.

What is the term with the highest power of x? It is 3x7-4x6. The exponent of x in this term is .

Similarly, in the polynomial q(y)= 5y6−4y2−6, the term with the highest power of y is 5y6-4y2 and the exponent of y in this term is .

We call the highest power of the variable in a polynomial as the degree of the polynomial. So, the degree of the polynomial 3x7-4x6+ x + 9 is 7 and the degree of the polynomial 5y6 -4y2-6. The degree of a non-zero constant polynomial is zero.

Now observe the polynomials p(x) = 4x+5, q(y) = 2y, r(t) = t + 2 and s(u) = 3−u. Do you see anything common among all of them?

The degree of each of these polynomials is .

A polynomial of degree one is called a linearquadratic polynomial.

Some more linear polynomials in one variable are 2x−1, 2y+1, 2−u. Now, try and find a linear polynomial in x with 3 terms? You would not be able to find it because a linear polynomial in x can have at most two terms.

Now consider the polynomials : 2x2 + 5, 5x2 + 3x , x2 and x2 + 25 x.

Do you agree that they are all of degree two? A polynomial of degree two is called a quadraticlinear polynomial.

We call a polynomial of degree threefour: a cubic polynomial. Some examples of a cubic polynomial in x are 4x3,2x3+1,5x3+x2,6x3-x,6-x3,2x3+4x2+6x+7

How many terms do you think a cubic polynomial in one variable can have?

It can have at most terms. These may be written in the form ax3+ bx2 + cx + d, where a ≠ 0 and a, b, c and d are constants.

Instructions

A polynomial function p(x) takes different values depending on the input x . To find the value of a polynomial at a given x , simply substitute that value into the expression.

Example

Consider the polynomial: p(x) = x2 - 2x - 3

To find the value of p(x) at x = 1 : p(1) = 12 - 2(1) - 3

= 1 - 2 - 3 =

Thus, the value of p(x) at x = 1 is .

Similarly, for x = 0 : p(0) = 02 - 2(0) - 3

= 0 - 0 - 3 =

So, p(0) = .

Zeroes of a Polynomial

Consider the polynomial p(x) = 5x3-2x2 + 3x-2

Instructions

Now, consider the polynomial p(x) = x – 1.

What is p(1)? Note that : p(1) = 1 – 1 = .

As p(1) = 0, we say that 1 is a zero of the polynomial p(x).

Similarly, you can check that 2 is a zero of q(x), where q(x) = x – 2.

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In general, we say that a zero of a polynomial p(x) is a number c such that p (c) = 0. In other words,

You must have observed that the zero of the polynomial x – 1 is obtained by equating it to 0, i.e., x – 1 = 0, which gives x = .

We say p(x) = 0 is a polynomial equation and 1 is the root of the polynomial equation p(x) = 0. So we say 1 is the zero of the polynomial x – 1, or a root of the polynomial equation x – 1 = 0.

Now, consider the constant polynomial 5. Can you tell what its zero is ? NoYes

It has no zero of polynomial because replacing x by any number in 5x0 still gives us 5.

In fact, a non-zero constant polynomial has no zero. What about the zeroes of the zero polynomial? By convention, every real number is a zero of the zero polynomial.

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