Degree of a Polynomial: Applications, Importance and Sample Questions

Polynomials are a significant concept of mathematics, and so is the degree of polynomials. The degree of polynomials helps in finding out the maximum number of solutions a function could include, and the number of times a function would cross the x-axis when graphed.

Polynomial is the highest exponential power of the variable in the polynomial equation. To check the degree of polynomials, only the variables are considered, and not the coefficients. In a polynomial, the degree of that polynomial is indicated by the highest exponential power of the variable term in the polynomial.

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For an nth degree polynomial function with real coefficients and where the variable is represented as x, having the highest power n, n takes whole number values. For example: 7x⁴ + 5x³+ 2.

The video below explains this:

Polynomials Detailed Video Explanation:

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How to Find the Degree of a Polynomial?

The polynomial that has the highest power of x is 3x⁵ and the highest exponent is therefore 5. Hence, it would be said that the highest degree of the polynomial above is 5. The degree of the polynomial can be represented by Deg(p(x)).

Degree of a Zero Polynomial

When all the coefficients of the variables are zero, the polynomial is said to be a zero polynomial. Thus, the degree of that polynomial is either in a negative way or is undefined (-1 or ∞).

Degree of a Constant Polynomial

A constant polynomial has no variables. There is no power on it since there is no exponent. Therefore, the degree of constant polynomial is zero, for instance, for 7 or 7x⁰, degree = 0.

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Degree of a Polynomial with More Than One Variable

The degree of a polynomial that has more than a single variable can be calculated by adding the exponential values of each variable in it. Example: 6x³ + 7x²y² + 3xy

Where 6x³ has a degree of 3, 7x²y² has a degree of 4 (x has an exponent of 2, y has 2, and 2+2=4), 3xy has a degree of 2 (x has an exponent of 1, y has 1, and 1+1=2).

The largest degree is 4 thus; the polynomial has a degree of 4.

On the basis of degree, the polynomials can be classified as the following:

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Applications

Following are the applications of the degree of the polynomial:

1. To find out the maximum number of solutions that a function could have.
2. To find out the most number of times that function will cross the x-axis when graphed.
3. To find out if the polynomial expression is homogeneous or not and if it determines the degree of each term.

When the degrees of the term are equal, the polynomial expression, in that case, is said to be homogeneous, and otherwise, it is said to be non-homogeneous. For example: in 5x³ + 2xy²+6y³ the degree of all the terms is 3, therefore, the given polynomial expression is homogeneous.

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How to Find the Degree of a Polynomial?

A Polynomial is a merging of variables assigned with exponential powers and coefficients. The steps to find the degree of a polynomial are as follows:- For example if the expression is :

2x⁵ + 4x³ + 3x⁵+ 5x²+ 7 + 9x + 4

• Step 1: Combine all the like terms that are the terms with the variable terms.

(3x⁵ + 2x⁵) + 4x³ + 5x²+ 9x + (7 +4)

• Step 2: Ignore all the coefficients,

x⁵+ x³+ x²+ x¹ + x⁰

• Step 3: Arrange the variable in descending order of their powers

x⁵+ x³+ x²+ x¹ + x⁰

• Step 4: The largest power of the variable is the degree of the polynomial

deg( x⁵+ x³+ x²+ x¹ + x⁰) = 5

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Importance of the Degree of a Polynomial

The degree of the terms in the polynomial plays an important role in determining whether the given polynomial expression is homogeneous or not.

The homogeneity of polynomial expression can be found by evaluating the degree of each term of the polynomial. For example, 2x³ + 3xy² + 5y³ is a multivariable polynomial. To check whether the polynomial expression is homogeneous, we need to determine the degree of each term. If all the degrees of the term are equal, then the polynomial expression is homogeneous and then the expression is non-homogenous if the degrees are not equal.

In the above example, the degree of all the terms is 3. Hence, the given example is a homogeneous polynomial of degree 3.

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Example Questions Using Degree of Polynomials Concept

Some of the examples of the polynomial with its degree are:

• 6x⁵ + 3x² – 2x + 3 – The degree of the polynomial is 5
• 16x³  – 3x² + 1 – The degree of the polynomial is 3
• 8x + 14 – The degree of the polynomial is 1
• 2 – The degree of the polynomial is 0

Example: Find the degree, constant and leading coefficient of the polynomial expression 2x² + 3x + 5.

Solution: Given Polynomial: 2x² + 3x + 5.

Here, the degree of the polynomial is 3, because the highest power of the variable of the polynomial is 3.

Constant is 3

Leading Coefficient is 2. Because the leading term of the polynomial is 2x²

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Tips and Tricks

The following tips and tricks could be followed in order to find the degree of any polynomial:

• Identify each term of the given polynomial.
• Combine all the like terms, the variable terms; overlook the constant terms.
• Arrange the terms in descending order of their exponential powers.
• The term with the highest exponent defines the degree of the polynomial.

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Points to Remember

1. Degree of a polynomial with one variable: the largest exponent of the variable.
2. The degree of any polynomial expression with a root such as 3√x is 1/2.
3. To find the degree of rational expression, the degree on the top (numerator) has to be subtracted with the degree of the bottom i.e. denominator.
4. The term “Polynomial '' comes from the word Poly, meaning many and nominal (here), meaning Term therefore, it means many terms.
5. A polynomial is made up of terms that can only be added, subtracted, or multiplied.
6. A quadratic polynomial in x with real coefficients is of the form ax² + bx + c, where a, b, c are real numbers with a ≠ 0.

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