DEGREE OF A POLYNOMIAL

The highest exponent of a variable in a polynomial is called the degree of the polynomial.

Polynomial is an algebraic expression which can be obtained by using mathematical operators like addition, subtraction, multiplication and division to combine variables and constants.

Consider the polynomial (2x + 3), which can be obtained by multiplying the variable x with the constant 2 and then adding the constant 3 to the product.

To know the degree of a polynomial, first we have to understand the degree of a variable by relating it with the exponents of numbers.

If you consider the square numbers, they have different bases with the same exponent 2. The geometrical representation of square numbers are shown below.

In general, if you consider the variable y as the length of a square, then its area is

= side x side

= (y x y)

= y² square units.

Therefore, we have an algebraic expression with exponent notation.

Consider the monomial y². The highest exponent of y² is 2, which is the degree of the monomial y².

Let us consider the polynomial : p³ + 5p² - 3.

The terms of the above polynommial are p³, 5p² and -3. Exponent of the term p³ is 3 and 5p² is 2. Thus, the term p³ has the highest exponent, that is 3.

Now, consider the polynomial, 5p⁴ - 3p³q² + 5pq + 3. 

Take each term and check its exponent. In 5p⁴, exponent is 4, hence its degree is 4. In -3p³q², the sum of exponents of p and q is 5, hence its degree is 5. In 8pq, the sum of exponents is 2. Therefore, the term with highest exponent in the above polynomial is -3p³q² and its power is 5, which is called as degree of this polynomial.

The term(s) containing the highest exponent of the variables in a polynomial is called the degree of polynomial.

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

Solved Examples

Examples 1-11 : In each case, find the degree of the given polynomial.

Example 1 :

y⁶

Solution :

In y⁶, the exponent is 6.

Thus, the degree of the polynomial is 6.

Example 2 :

-5y³z²

Solution :

In -5y³z², the sum of exponents of y and z is 5

That is,

3 + 2 = 5

Thus, the degree of the polynomial is 5.

Example 3 :

3pq²r³

Solution :

In 3pq²r³, the sum of exponents of p, q and r is 6.

That is,

1 + 2 + 3 = 6

Thus, the degree of the polynomial is 6.

Example 4 :

4x - 5

Solution :

The terms of the given polynomial are 4x, and -5.

Exponent of each of the terms : 1, 0.

Terms with highest exponent : 4x¹.

Therefore, degree of the polynomial is 1.

Example 5 :

5x - 17x⁶

Solution :

The terms of the given polynomial are 5x and -17x⁶.

Exponent of each of the terms : 0, 6.

Terms with highest exponent : -17x⁶.

Therefore, degree of the polynomial is 6.

Example 6 :

-6x⁴ - 5x - 9x³

Solution :

The terms of the given polynomial are -6x⁴, -5x and -9x³.

Exponent of each of the terms : 4, 1, 3.

Terms with highest exponent : -6x⁴.

Therefore, degree of the polynomial is 4.

Example 7 :

x² - x + y³

Solution :

The terms of the given polynomial are x², -x and y³.

Exponent of each of the terms : 2, 1, 3.

Terms with highest exponent : y³.

Therefore, degree of the polynomial is 3.

Example 8 :

7x⁴ + 9x³ - x + 3

Solution :

The terms of the given polynomial are 7x⁴, 9x³, -x and 3.

Exponent of each of the terms : 4, 3, 1, 0.

Terms with highest exponent : 7x⁴.

Therefore, degree of the polynomial is 4.

Example 9 :

0.3xy² + 0.25xy + 0.78

Solution :

The terms of the given polynomial are 0.3xy², 0.25xy and 0.78.

Exponent of each of the terms : 3, 2, 1.

Terms with highest exponent : 0.3xy².

Therefore, degree of the polynomial is 3.

Example 10 :

0.75a²b - 2a³b⁵

Solution :

The terms of the given polynomial are 0.75a²b and -2a³b⁵.

Exponent of each of the terms : 3, 8.

Terms with highest exponent : - 2a³b⁵.

Therefore, degree of the polynomial is 8.

Example 11 :

x³y² + x²y³ - x⁴ + 5

Solution :

The terms of the given polynomial are x³y², x²y³, -x⁴ and  5.

Exponent of each of the terms : 5, 5, 4, 0.

Terms with highest exponent : x³y², x²y³.

Therefore, degree of the polynomial is 5.

Example 12 :

Add the following two polynomials and find the degree.

(3p² + 4pq + 7q²) and (5p² - 6pq + 7q²)

Solution :

= (3p² + 4pq + 7q²) + (5p² - 6pq + 7q²)

= 3p² + 4pq + 7q² + 5p² - 6pq + 7q²

Group like terms together.

= (3p² + 5p²) + (4pq - 6pq) + (7q² + 7q²)

= 8p² + (-2pq) + 14q²

= 8p² - 2pq + 14q²

Thus, the degree of the polynomial is 2.

Example 13 :

Subtract (y⁴ - y² + y + 3) from (3y⁴ - 2y² - 7y + 6) and find the degree.

Solution :

= (3y⁴ - 2y² - 7y + 6) - (y⁴ - y² + y + 3)

Distributive the negative sign.

= 3y⁴ - 2y² - 7y + 6 - y⁴ + y² - y - 3

Group like terms together.

= (3y⁴ - y⁴) + (-2y² + y²) + (-7y - y) + (6 - 3)

Combine like terms.

= 2y⁴ + (-y²) + (-8y) + 3

= 2y⁴ - y² - 8y + 3

Hence, the degree of the polynomial is 4.

Example 14 :

Simplify and find the degree of the resulting polynomial.

(4x² + 3y) - (3x + 9y²) - (3x² - 6y²) + (5x - y)

Solution :

= (4x² + 3y) - (3x + 9y²) - (3x² - 6y²) + (5x - y)

= 4x² + 3y - 3x - 9y² - 3x² + 6y² + 5x - y

Group like terms together.

= (4x² - 3x²) + (-9y² + 6y²) + (-3x + 5x) + (3y - y)

Combine like terms.

= x² + (-3y²) + 2x + 2y

= x² - 3y² + 2x + 2y

Hence, the degree of the polynomial is 2.

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