DEGREE OF AN ALGEBRAIC EXPRESSION

Let us briefly recap the construction of an algebraic expression. To construct an algebraic expression, we use mathematical operators like addition, subtraction, multiplication and division to combine variables and constants.

Consider the algebraic expression (5y + 7), which can be obtained by multiplying the variable y with the constant 5 and then adding the constant 7 to the product.

To know the degree of an expression, first let us try to understand the degree of a variable by relating it with the exponents of numbers.

Let us consider the square numbers. They have different base and same exponents. The geometrical representation of square numbers are given below.

degreeofanexpression1

In general, if we consider the side of a square as a variable ‘y’ then its area will be (y x y) sq. units. This can be denoted as ‘y2’. Thus we have an algebraic expression with exponent notation.

degreeofanexpression2

If we consider the term x2 as a monomial expression, the highest power of the expression is its exponent, that is 2.
Similarly, when length l units and width w units are variables of a rectangle, then its area is

l x w = lw sq. units.

We can consider lw as a term in an algebraic expression,  where l and w are factors of lw. The highest power of the expression lw is also 2, as we have to add up the powers of the variable factors.

Note

(i) When no exponent is explicitly shown in a variable of a term, it is understood to be 1.

For example, 11p = 11p1.

(ii) For an expression in x, if the terms of the expressions are in descending powers of x and the like terms are added, then we say that it is in the standard form.

For example, x4 - 3x3 + 5x2 - 7x + 9 is in the standard form. It is easier to find the highest power term when the expression is in standard form. Highest power of this expression is 4.

(iii) The highest degree term of an algebraic expression is called as leading term.

Let us consider an algebraic expression : x3 - 3x2 + 4.

The terms of the above expression are x3, -3x2 and 4. Exponent of the term xis 3 and -3x2 is 2. Thus, the term x3 has the highest exponent, that is 3. 

Now, consider the expression, 3x4 - 4x3y2 + 8xy + 7. 

Take each term and check its power. In 3x4, exponent is 4, hence its degree is 4. In -4x3y2, the sum of powers of x and y is 5, hence its degree is 5. In 8xy, the sum of powers is 2. Therefore, the term with highest power in the above expression is -4x3y2 and its power is 5, which is called as degree of this expression.

The term(s) containing the highest power of the variables in an expression is called the degree of expression.

The degree of any term in an expression can only be a positive integer. Also, degree of expression doesn’t depend on the number of terms, but on the power of variables in the individual terms. The degree of constant term is 0.

Try These

1. Complete the following table:

degreeofanexpression3

2. Identify the like terms from the following :

(i) 2x2y, 2xy2, 3xy2, 14x2y, 7yx
(ii) 3x2y2, y3x, y3x2, -y3x, 3y3x
(iii) 11pq, -pq, 11pqr, -11pq, pq

Solved Problems

Problems 1-5 : In each case, find the degree of the given expression.

Problem 1 :

x5

Solution :

In x5, the exponent is 5. Thus, the degree of the expression is 5.

Problem 2 :

-3p3q2

Solution :

In -3p3q2, the sum of powers of p and q is 5

That is,

3 + 2 = 5

Thus, the degree of the expression is 5.

Problem 3 :

-4xy2z3

Solution :

In -4xy2z3, the sum of powers of x, y and z is 6.

That is,

1 + 2 + 3 = 6

Thus, the degree of the expression is 6.

Problem 4 :

12xyz - 3x3y2z + z8

Solution :

The terms of the given expression are 12xyz, -3x3y2z and z8.

Degree of each of the terms : 3, 6, 8.

Terms with highest degree : z8.

Therefore, degree of the expression is 8.

Problem 5 :

3a3b4 - 16c6 + 9b2c5 + 7 

Solution :

The terms of the given expression are 3a3b4, -16c6, 9b2c5, 7.

Degree of each of the terms : 7, 6, 7, 0.

Terms with highest degree : 3a3b49b2c5

Therefore, degree of the expression is 7.

Problem 6 :

Add the expressions 4x+ 3xy + 9yand 2x- 9xy + 6yand find the degree.

Solution :

= (4x+ 3xy + 9y2) + (2x- 9xy + 6y2)

= 4x+ 3xy + 9y2 + 2x- 9xy + 6y2

Group like terms together.

= (4x+ 2x2) + (3xy - 9xy) + (9y2 + 6y2)

= 6x2 + (-3xy) + 15y2

= 6x2 - 3xy + 15y2

Thus, the degree of the expression is 2.

Problem 7 :

Subtract (x- x+ x + 3) from (3x- 2x- 7x + 6) and find the degree.

Solution :

= (3x- 2x- 7x + 6) - (x- x+ x + 3)

Distributive the negative sign.

= 3x- 2x- 7x + 6 - x+ x- x - 3

Group like terms together.

= (3x- x3) + (-2x2 + x2) + (-7x - x) + (6 - 3)

Combine like terms.

= 2x3 + (-x2) + (-8x) + 3

= 2x3 - x2 - 8x + 3

Hence, the degree of the expression is 3.

Problem 8 :

Simplify and find the degree of the resulting expression.

(4m+ 3n) - (3m + 9n2) - (3m- 6n2) + (5m - n)

Solution :

= (4m+ 3n) - (3m + 9n2) - (3m- 6n2) + (5m - n)

= 4m+ 3n - 3m - 9n2 - 3m+ 6n2 + 5m - n  

Group like terms together.

= (4m- 3m2) + (-9n2 + 6n2) + (-3m + 5m) + (3n - n)

Combine like terms.

= m2 + (-3n2) + 2m + 2n

= m2 - 3n2 + 2m + 2n

Hence, the degree of the expression is 2.

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