A polynomial is an expression that shows sums and differences of multiple terms made of coefficients and variables.
A polynomial expression with zero degree is called a constant. A polynomial expression with a degree of one is called linear. A polynomial expression with degree two is called quadratic, and a polynomial with degree three is called cubic.
Degree of Polynomials with One Variable
The degree of a polynomial with one variable is the value of the largest exponent.
For example, the degree of the polynomial expression \(3x^5+4x-2\), is 5 because of the term \(3x^5\) that has an exponent of 5.
Here are some additional examples of polynomial expressions and the degree of the expression:
| Example | Degree |
|---|---|
| \(12\) | \(0\) |
| \(3x-2\) | \(1\) |
| \(5x^2-x+7\) | \(2\) |
| \(x^3+2x^2-4x-12\) | \(3\) |
| \(2x^4-4x^3+x^2-3x-3\) | \(4\) |
Degree of Polynomials with Multiple Variables
The degree of a polynomial expression with multiple variables is the degree of the term with the largest degree, which can be calculated by adding the values of the exponents of the variables in that term.
For example, the degree of the term \(6x^2y^3\) is 5 because the exponent of \(x\) is 2 and the exponent of \(y\) is 3 and \(2+3=5\).
To find the degree of the polynomial expression \(3x^5 y^3-4x^4 y^2+x^2 y^3-2xy\), we start by finding the degree of each term:
- The term \(3x^5y^3\) has a degree of 8 because the exponent for \(x\) is 5 and for \(y\) is 3 and \(5+3=8\).
- The term \(-4x^4y^2\) has a degree of 6 because the exponent for \(x\) is 4 and for \(y\) is 2 and \(4+2=6\).
- The term \(x^2y^3\) has a degree of 5 because the exponent for \(x\) is 2 and for \(y\) is 3 and \(2+3=5\).
- The term \(-2xy\) has a degree of 2 because the exponent for \(x\) is 1 and for \(y\) is 1 and \(1+1=2\).
Therefore, the degree of the polynomial expression, \(3x^5y^3-4x^4y^2+x^2y^3-2xy\), is 8 because that is the highest degree of one of the terms.
Degree of Polynomials in a Fraction
The degree of a polynomial expression in fraction form is the degree of the expression in the numerator minus the degree of the expression in the denominator.
For example, the degree of the fraction \(\frac{2x^3y^2-3x^6y}{x^3y^3+2x^2y^2}\) is 1 because the degree of the numerator is 7 and the degree of the denominator is 6 and \(7–6=1\).
Degree of Polynomial Practice Problems
Here are a few sample questions going over the degree of polynomials.
What is the degree of the polynomial expression \(4x^4y^3-5x^3y^4+3x^2y^4+5x^3y^4+x^2y^2-yx-8\)?
5
6
7
8
We will start by checking to see if there are any terms that are the same and can be combined. Since the terms \(-5x^3y^4\) and \(5x^3y^4\) have the same variables, they can be combined, which in this case adds up to 0. We will use the remaining expression, \(4x^4y^3+3x^2y^4+x^2y^2-yx-8\), to find the degree by finding the degree of each term first. The term \(4x^4y^3\) has a degree of 7. The term \(3x^2y^4\) has a degree of 6. The term \(x^2y^2\) has a degree of 4. The term \(-yx\) has a degree of 2. And the term -8 has a degree of 0. Therefore, the degree of the polynomial expression is 7 because 7 is the highest degree from this list.
What is the degree of the polynomial expression \(3xy^4+2x^2y^2-8x^3y^6+4x4y-y^5\)?
5
6
8
9
Check if there are any terms that can be combined. In this case there are none. We will find the degree of each term. The degree of the first term, \(3xy^4\), is 5. The degree of the second term, \(2x^2y^2\), is 4. The degree of the third term, \(-8x^3y^6\), is 9. The degree of the fourth term, \(4x^4y\), is 5. And the degree of the fifth term, \(-y^5\), is 5. Since the largest degree is 9, the degree of the polynomial expression is 9.
What is the degree of the polynomial expression \((x^2+x-3)(y^2-y+2)\)?
3
4
5
6
We will start by using the distributive property to expand and simplify the polynomial expression which becomes \(x^2y^2-x^2y+2x^2+xy^2-xy+2x-3y^2+3y-6\). Now we will find the degree of each term. The degree of \(x^2y^2\) is 4, degree of \(-x^2y\) is 3, degree of \(2x^2\) is 2, degree of \(xy^2\) is 3, degree of \(-xy\) is 2, degree of \(2x\) is 1, degree of \(-3y^2\) is 2, degree of \(3y\) is 1 and degree of -6 is 0. Therefore, since the largest degree is 4, the degree of the polynomial expression is 4.
What is the degree of the polynomial expression \(\frac{6x^5-7x^2y^2}{2x^2-2x^3+2xy}\)?
2
3
4
6
To find the degree of a polynomial expression in fraction form, we find the degree of the polynomial in the numerator and the degree of the polynomial in the denominator and subtract the two numbers. The degree of the polynomial in the numerator is 5, the degree of the polynomial in the denominator is 3. \(5–3=2\); therefore, the degree of the polynomial expression is 2.
What is the degree of the polynomial expression \(\frac{-3x^7+6x^5}{2x^2-x}\)?
4
5
6
9
The degree of the polynomial in the numerator is 7. The degree of the polynomial in the denominator is 2. To find the degree of the polynomial fraction in fraction form, we will subtract the degree of the denominator from the degree of the numerator, \(7–2=5\); therefore, the degree of the polynomial expression is 5.
