Difference between revisions of "Find the Degree of a Polynomial"
(importing article from wikihow) |
m (Update ref tag) |
||
| Line 1: | Line 1: | ||
| − | Polynomial means "many terms," and it can refer to a variety of expressions that can include constants, variables, and exponents. For example, ''x - 2'' is a polynomial; so is ''25.'' To find the degree of a polynomial, all you have to do is find the largest exponent in the polynomial.<ref>http://www.mathwarehouse.com/algebra/polynomial/degree-of-polynomial.php</ref> If you want to find the degree of a polynomial in a variety of situations, just follow these steps. | + | Polynomial means "many terms," and it can refer to a variety of expressions that can include constants, variables, and exponents. For example, ''x - 2'' is a polynomial; so is ''25.'' To find the degree of a polynomial, all you have to do is find the largest exponent in the polynomial.<ref name="rf1">http://www.mathwarehouse.com/algebra/polynomial/degree-of-polynomial.php</ref> If you want to find the degree of a polynomial in a variety of situations, just follow these steps. |
[[Category:Algebra]] | [[Category:Algebra]] | ||
| Line 8: | Line 8: | ||
#Drop all of the constants and coefficients. The constant terms are all of the terms that are not attached to a variable, such as 3 or 5. The coefficients are the terms that ''are'' attached to the variable. When you're looking for the degree of a polynomial, you can either just actively ignore these terms or cross them off. For instance, the coefficient of the term 5x<sup>2</sup> would be 5. The degree is independent of the coefficients, so you don't need them. | #Drop all of the constants and coefficients. The constant terms are all of the terms that are not attached to a variable, such as 3 or 5. The coefficients are the terms that ''are'' attached to the variable. When you're looking for the degree of a polynomial, you can either just actively ignore these terms or cross them off. For instance, the coefficient of the term 5x<sup>2</sup> would be 5. The degree is independent of the coefficients, so you don't need them. | ||
#*Working with the equation 5x<sup>2</sup> - 3x<sup>4</sup> - 5 + x, you would drop the constants and coefficients to get x<sup>2</sup> - x<sup>4</sup> + x. | #*Working with the equation 5x<sup>2</sup> - 3x<sup>4</sup> - 5 + x, you would drop the constants and coefficients to get x<sup>2</sup> - x<sup>4</sup> + x. | ||
| − | #Put the terms in decreasing order of their exponents. This is also called putting the polynomial in ''standard form.''<ref>http://www.mathsisfun.com/algebra/polynomials.html</ref>. The term with the highest exponent should be first, and the term with the lowest exponent should be last. This will help you see which term has the exponent with the largest value. In the previous example, you would be left with <br>-x<sup>4</sup> + x<sup>2</sup> + x. | + | #Put the terms in decreasing order of their exponents. This is also called putting the polynomial in ''standard form.''<ref name="rf2">http://www.mathsisfun.com/algebra/polynomials.html</ref>. The term with the highest exponent should be first, and the term with the lowest exponent should be last. This will help you see which term has the exponent with the largest value. In the previous example, you would be left with <br>-x<sup>4</sup> + x<sup>2</sup> + x. |
#Find the power of the largest term. The power is simply number in the exponent. In the example, -x<sup>4</sup> + x<sup>2</sup> + x, the power of the first term is 4. Since you've arranged the polynomial to put the largest exponent first, that will be where you will find the largest term. | #Find the power of the largest term. The power is simply number in the exponent. In the example, -x<sup>4</sup> + x<sup>2</sup> + x, the power of the first term is 4. Since you've arranged the polynomial to put the largest exponent first, that will be where you will find the largest term. | ||
| − | #Identify this number as the degree of the polynomial. You can just write that the degree of the polynomial = 4, or you can write the answer in a more appropriate form: ''deg (3x<sup>2</sup> - 3x<sup>4</sup> - 5 + 2x + 2x<sup>2</sup> - x) = 3.'' You're all done.<ref | + | #Identify this number as the degree of the polynomial. You can just write that the degree of the polynomial = 4, or you can write the answer in a more appropriate form: ''deg (3x<sup>2</sup> - 3x<sup>4</sup> - 5 + 2x + 2x<sup>2</sup> - x) = 3.'' You're all done.<ref name="rf1" /> |
#Know that the degree of a constant is zero. If your polynomial is only a constant, such as 15 or 55, then the degree of that polynomial is really zero. You can think of the constant term as being attached to a variable to the degree of 0, which is really 1. For example, if you have the constant 15, you can think of it as ''15x<sup>0</sup>,'' which is really 15 x 1, or 15. This proves that the degree of a constant is 0. | #Know that the degree of a constant is zero. If your polynomial is only a constant, such as 15 or 55, then the degree of that polynomial is really zero. You can think of the constant term as being attached to a variable to the degree of 0, which is really 1. For example, if you have the constant 15, you can think of it as ''15x<sup>0</sup>,'' which is really 15 x 1, or 15. This proves that the degree of a constant is 0. | ||
=== a Polynomial with Multiple Variables === | === a Polynomial with Multiple Variables === | ||
#Write the expression. Finding the degree of a polynomial with multiple variables is only a little bit trickier than finding the degree of a polynomial with one variable. Let's say you're working with the following expression: | #Write the expression. Finding the degree of a polynomial with multiple variables is only a little bit trickier than finding the degree of a polynomial with one variable. Let's say you're working with the following expression: | ||
#*x<sup>5</sup>y<sup>3</sup>z + 2xy<sup>3</sup> + 4x<sup>2</sup>yz<sup>2</sup> | #*x<sup>5</sup>y<sup>3</sup>z + 2xy<sup>3</sup> + 4x<sup>2</sup>yz<sup>2</sup> | ||
| − | #Add the degree of variables in each term. Just add up the degrees of the variables in each of the terms; it does not matter that they are different variables. Remember that the degree of a variable without a written degree, such as x or y, is just one. Here's how you do it for all three terms:<ref>http://www.mathsisfun.com/algebra/degree-expression.html</ref> | + | #Add the degree of variables in each term. Just add up the degrees of the variables in each of the terms; it does not matter that they are different variables. Remember that the degree of a variable without a written degree, such as x or y, is just one. Here's how you do it for all three terms:<ref name="rf3">http://www.mathsisfun.com/algebra/degree-expression.html</ref> |
#*x<sup>5</sup>y<sup>3</sup>z = 5 + 3 + 1 = 9 | #*x<sup>5</sup>y<sup>3</sup>z = 5 + 3 + 1 = 9 | ||
#*2xy<sup>3</sup> = 1 + 3 = 4 | #*2xy<sup>3</sup> = 1 + 3 = 4 | ||
| Line 22: | Line 22: | ||
#Identify this number as the degree of the polynomial. 9 is the degree of the entire polynomial. You can write the final answer like this: ''deg (x<sup>5</sup>y<sup>3</sup>z + 2xy<sup>3</sup> + 4x<sup>2</sup>yz<sup>2</sup>) = 9''. | #Identify this number as the degree of the polynomial. 9 is the degree of the entire polynomial. You can write the final answer like this: ''deg (x<sup>5</sup>y<sup>3</sup>z + 2xy<sup>3</sup> + 4x<sup>2</sup>yz<sup>2</sup>) = 9''. | ||
=== a Rational Expression=== | === a Rational Expression=== | ||
| − | #Write down the expression. Let's say you're working with the following expression: (x<sup>2</sup> + 1)/(6x -2).<ref | + | #Write down the expression. Let's say you're working with the following expression: (x<sup>2</sup> + 1)/(6x -2).<ref name="rf3" /> |
#Eliminate all coefficients and constants. You won't need the coefficients or constant terms to find the degree of a polynomial with fractions. So, eliminate the 1 from the numerator and the 6 and -2 from the denominator. You're left with x<sup>2</sup>/x. | #Eliminate all coefficients and constants. You won't need the coefficients or constant terms to find the degree of a polynomial with fractions. So, eliminate the 1 from the numerator and the 6 and -2 from the denominator. You're left with x<sup>2</sup>/x. | ||
#Subtract the degree of the variable in the denominator from the degree of the variable in the numerator. The degree of the variable in the numerator is 2 and the degree of the variable in the denominator is 1. So, subtract 1 from 2. 2-1 = 1. | #Subtract the degree of the variable in the denominator from the degree of the variable in the numerator. The degree of the variable in the numerator is 2 and the degree of the variable in the denominator is 1. So, subtract 1 from 2. 2-1 = 1. | ||
