Polynomials in One Variable

Let us begin by recalling that a variable is denoted by a symbol that can take any real value. We use the letters x, y, z, etc. to denote variables. Notice that 2x, 3x, – x, -12 x are algebraic expressions.

All these expressions are of the form (a constant) × x.

Now suppose we want to write an expression which is (a constant) × (a variable) and we do not know what the constant is. In such cases, we write the constant as a, b, c, etc. So the expression will be ax, say.

However, there is a difference between a letter denoting a constant and a letter denoting a variable. The values of the constants remain the samechange throughout a particular situation, that is, the values of the constants do not change in a given problem, but the value of a variable can keep changing.

Now, consider a square of side 3 units. What is its perimeter? You know that the perimeter of a square is the sum of the lengths of its fourthree sides.

Here, each side is 3 units. So, its perimeter is 4 × 3, i.e., units.

What will be the perimeter if each side of the square is 10 units?

The perimeter is 4 × 10, i.e., 40 units.

In case the length of each side is x units (see Fig. 2.2), the perimeter is given by 4x units. So, as the length of the side varies, the perimeter varies.

Can you find the area of the square PQRS? It is x × x = x2x3 square units.

x2 is an algebraic expression. You are also familiar with other algebraic expressions like 2x, x2+2x , x3−x2+4x+7. Note that, all the algebraic expressions we have considered so far have only whole numbers as the exponents of the variable. Expressions of this form are called polynomials in one variable.

In the examples above, the variable is x. For instance, x3-x2+4x+7 is a polynomial in x. Similarly, 3y2+5y is a polynomial in the variable y and t2+ 4 is a polynomial in the variable t.

In the polynomial x2 + 2x, the expressions x2 and 2x are called the terms of the polynomial.

Similarly, the polynomial 3y2+5y+7 has threetwo terms, namely, 3y2,5y and 7.

Can you write the terms of the polynomial −x3+4x2+7x−2? This polynomial has terms, namely, −x3, 4x2, 7x and –2.

Each term of a polynomial has a coefficient. So, in −x3+4x2+7x−2 the coefficient of x3 is –13, the coefficient of x2 is 42, the coefficient of x is 71 and –2 is the coefficient of x0 . Do you know the coefficient of x in x2−x+7 ? It is –12.

Let us consider these examples: −x3+ 4x2+7x-2, x3-x2 + 4x + 7

In fact, -2, 7 are examples of constant polynomials. The constant polynomial 0 is called the zero polynomial. This plays a very important role in the collection of all polynomials, as you will see in the higher classes.

Example : x2+ 2x.

Drop the zero and nonzero polynomials into the concerned boxes.

Instructions

Now, consider algebraic expressions such as x + 1x , + 3, + y2 . Do you know that you can write x + 1x = x + x−1 ? Here, the exponent of the second term, i.e., x−1 is –1, which is notis a whole number. So, this algebraic expression is notis a polynomial.

A polynomial can have any (finite) number of terms. For instance, x150 + x149 + ---- + x2+ x + 1 is a polynomial with 151 terms.

Consider the polynomials:

2x,2,5x3,−5x2,y and u4

Do you see that each of these polynomials has only one term ? Polynomials having only one term are called monomials (‘mono’ means ‘one’).

Now observe each of the following polynomials:

p(x) = x+1,

q(x) = x2−x,

r(y) = y9+1,

t(u) = u15−u2

How many terms are there in each of these? Each of these polynomials has only 23 terms. Polynomials having only terms are called binomials (‘bi’ means ‘two’).

Similarly, polynomials having only terms are called trinomials (‘tri’ means ‘three’). Some examples of trinomials are:

Drop the monomials,binomials and trinomials into the concerned boxes.

Instructions

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

Instructions

Now, that you have seen what a polynomial of degree 1, degree 2, or degree 3 looks like, can you write down a polynomial in one variable of degree n for any natural number n? A polynomial in one variable x of degree n is an expression of the form:

anxn + an−1xn−1 + ......... + a1x+ a0 = 0

So far we have dealt with polynomials in one variable only. We can also have polynomials in more than one variable. For example, x2+ y2 + xyz (where variables are x, y and z) is a polynomial in three variables. Similarly p2 + q10 + r (where the variables are p, q and r), u3 + v2 (where the variables are u and v) are polynomials in three and two variables, respectively. You will be studying such polynomials in detail later.