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, -1/2 x` are algebraic expressions. All these expressions are of the form`text ( (a constant) ) × x`. Now suppose we want to write an expression which is`text ( (a constant) ) × text ( (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 same 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 (see Fig. 2.1). What is its perimeter? You know that the perimeter of a square is the sum of the lengths of its four sides.
Here, each side is 3 units. So, its perimeter is `4 × 3`, i.e., 12 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 = x^2` square units. `x^2` is an algebraic expression. You are also familiar with other algebraic expressions like `2x, x^2 + 2x, x^3 – x^2 + 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, `x^3 – x^2 + 4x + 7` is a polynomial in `x`. Similarly, `3y^2 + 5y` is a polynomial in the variable` y` and `t^2 + 4` is a polynomial in the variable `t`.
In the polynomial `x^2 + 2x`, the expressions `x^2` and `2x` are called the terms of the polynomial. Similarly, the polynomial `3y^2 + 5y + 7` has three terms, namely, `3y^2, 5y `and 7.
Can you write the terms of the polynomial `–x^3 + 4x^2 + 7x – 2 ?` This polynomial has
`4` terms, namely, `–x^3, 4x^2, 7x` and `–2`.
Each term of a polynomial has a coefficient. So, in `–x^3 + 4x^2 + 7x – 2`, the coefficient of `x^3` is `–1`, the coefficient of `x^2` is 4, the coefficient of `x` is `7` and `–2` is the coefficient of `x^0` (Remember, `x^0 = 1`). Do you know the coefficient of `x` in `x^2 – x + 7?` It is `–1`.
`2` is also a polynomial. In fact, `2, –5, 7`, etc. 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.
Now, consider algebraic expressions such as `x + 1/x , sqrt (x ) + 3` and `root 3 (y) + y^2`. Do you know that you can write `x +1/x =x + x^(-1)` ?Here, the exponent of the second term, i.e., ` x^(-1) ` is `-1` , which is not a whole number. So, this algebraic expression is not a polynomial.
Again, `sqrt (x) +3` can be written as `x^(1/2) + 3` . Here the exponent of x is `1/2` , which is not a whole number. So, is `sqrt (x) +3` a polynomial? No, it is not. What about `root 3 (y) + y^2 ?` It is also not a polynomial (Why?).
If the variable in a polynomial is x, we may denote the polynomial by `p(x)`, or `q(x)`, or `r(x)`, etc. So, for example, we may write :
` color { red} { p(x) = 2x^2 + 5x – 3 } `
` color {green } { q(x) = x^3 –1 } `
` color { blue } { r(y) = y^3 + y + 1 } `
` color { orange } { s(u) = 2 – u – u^2 + 6u^5 } `
A polynomial can have any (finite) number of terms. For instance, `x^150 + x^149 + ...+ x^2 + x + 1` is a polynomial with `151` terms.
Consider the polynomials` 2x, 2, 5x^3, –5x^2, y` and `u^4`. 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) = x^2 – x, r(y) = y^30 + 1, t(u) = u^43 – u^2`
How many terms are there in each of these? Each of these polynomials has only two terms. Polynomials having only two terms are called binomials (‘bi’ means ‘two’).
Similarly, polynomials having only three terms are called trinomials (‘tri’ means ‘three’). Some examples of trinomials are
` color { green } { p(x) = x + x^2 + π } `,
` color {red } { q(x) = sqrt 2 + x – x^2 } `,
` color { blue } { r(u) = u + u^2 – 2 } `,
` color { orange } { t(y) = y^4 + y + 5 } `.
Now, look at the polynomial` p(x) = 3x^7 – 4x^6 + x + 9`. What is the term with the highest power of `x` ? It is `3x^7`. The exponent of `x` in this term is `7`.
Similarly, in the polynomial `q(y) = 5y^6 – 4y^2 – 6`, the term with the highest power of `y` is `5y^6` and the exponent of y in this term is `6`. We call the highest power of the variable in a polynomial as the degree of the polynomial.
So, the degree of the polynomial `3x^7 – 4x^6 + x + 9` is `7` and the degree of the polynomial `5y^6 – 4y^2 – 6` is `6`. The degree of a non-zero constant polynomial is zero.
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, -1/2 x` are algebraic expressions. All these expressions are of the form`text ( (a constant) ) × x`. Now suppose we want to write an expression which is`text ( (a constant) ) × text ( (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 same 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 (see Fig. 2.1). What is its perimeter? You know that the perimeter of a square is the sum of the lengths of its four sides.
Here, each side is 3 units. So, its perimeter is `4 × 3`, i.e., 12 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 = x^2` square units. `x^2` is an algebraic expression. You are also familiar with other algebraic expressions like `2x, x^2 + 2x, x^3 – x^2 + 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, `x^3 – x^2 + 4x + 7` is a polynomial in `x`. Similarly, `3y^2 + 5y` is a polynomial in the variable` y` and `t^2 + 4` is a polynomial in the variable `t`.
In the polynomial `x^2 + 2x`, the expressions `x^2` and `2x` are called the terms of the polynomial. Similarly, the polynomial `3y^2 + 5y + 7` has three terms, namely, `3y^2, 5y `and 7.
Can you write the terms of the polynomial `–x^3 + 4x^2 + 7x – 2 ?` This polynomial has
`4` terms, namely, `–x^3, 4x^2, 7x` and `–2`.
Each term of a polynomial has a coefficient. So, in `–x^3 + 4x^2 + 7x – 2`, the coefficient of `x^3` is `–1`, the coefficient of `x^2` is 4, the coefficient of `x` is `7` and `–2` is the coefficient of `x^0` (Remember, `x^0 = 1`). Do you know the coefficient of `x` in `x^2 – x + 7?` It is `–1`.
`2` is also a polynomial. In fact, `2, –5, 7`, etc. 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.
Now, consider algebraic expressions such as `x + 1/x , sqrt (x ) + 3` and `root 3 (y) + y^2`. Do you know that you can write `x +1/x =x + x^(-1)` ?Here, the exponent of the second term, i.e., ` x^(-1) ` is `-1` , which is not a whole number. So, this algebraic expression is not a polynomial.
Again, `sqrt (x) +3` can be written as `x^(1/2) + 3` . Here the exponent of x is `1/2` , which is not a whole number. So, is `sqrt (x) +3` a polynomial? No, it is not. What about `root 3 (y) + y^2 ?` It is also not a polynomial (Why?).
If the variable in a polynomial is x, we may denote the polynomial by `p(x)`, or `q(x)`, or `r(x)`, etc. So, for example, we may write :
` color { red} { p(x) = 2x^2 + 5x – 3 } `
` color {green } { q(x) = x^3 –1 } `
` color { blue } { r(y) = y^3 + y + 1 } `
` color { orange } { s(u) = 2 – u – u^2 + 6u^5 } `
A polynomial can have any (finite) number of terms. For instance, `x^150 + x^149 + ...+ x^2 + x + 1` is a polynomial with `151` terms.
Consider the polynomials` 2x, 2, 5x^3, –5x^2, y` and `u^4`. 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) = x^2 – x, r(y) = y^30 + 1, t(u) = u^43 – u^2`
How many terms are there in each of these? Each of these polynomials has only two terms. Polynomials having only two terms are called binomials (‘bi’ means ‘two’).
Similarly, polynomials having only three terms are called trinomials (‘tri’ means ‘three’). Some examples of trinomials are
` color { green } { p(x) = x + x^2 + π } `,
` color {red } { q(x) = sqrt 2 + x – x^2 } `,
` color { blue } { r(u) = u + u^2 – 2 } `,
` color { orange } { t(y) = y^4 + y + 5 } `.
Now, look at the polynomial` p(x) = 3x^7 – 4x^6 + x + 9`. What is the term with the highest power of `x` ? It is `3x^7`. The exponent of `x` in this term is `7`.
Similarly, in the polynomial `q(y) = 5y^6 – 4y^2 – 6`, the term with the highest power of `y` is `5y^6` and the exponent of y in this term is `6`. We call the highest power of the variable in a polynomial as the degree of the polynomial.
So, the degree of the polynomial `3x^7 – 4x^6 + x + 9` is `7` and the degree of the polynomial `5y^6 – 4y^2 – 6` is `6`. The degree of a non-zero constant polynomial is zero.
