Find the Degree 7p^5-p/10+p^4
The problem is asking for the calculation of the degree of a given polynomial. In algebra, the degree of a polynomial is the highest power of the variable that appears in the polynomial. The polynomial in question is "7p^5 - p/10 + p^4." You are supposed to determine which term has the highest exponent and report that exponent as the degree of the polynomial.
$7 p^{5} - \frac{p}{10} + p^{4}$
Answer
Solution:
Step 1:
Determine the powers of the variable \( p \) in the polynomial's terms. The degree of a term is the sum of the exponents of the variables it contains.
For \( 7p^5 \), the degree is \( 5 \).
For \( -\frac{p}{10} \), the degree is \( 1 \) since \( p = p^1 \).
For \( p^4 \), the degree is \( 4 \).
Step 2:
Identify the highest degree among the terms to establish the polynomial's overall degree. In this case, the highest exponent is \( 5 \), which means the degree of the polynomial is \( 5 \).
Knowledge Notes:
The degree of a polynomial is the highest power of the variable in the polynomial. When determining the degree, one must look at each term individually. A term is a product of a constant and a variable raised to an exponent. The exponent can be a whole number, and in the case where a variable is not accompanied by an exponent, it is understood to be 1 (e.g., \( p \) is the same as \( p^1 \)).
When identifying the degree of a polynomial:
First, examine each term to find the exponent of the variable. If there are multiple variables in a term, add their exponents to find the degree of that term.
Compare the degrees of all the terms and select the largest one. This is the degree of the entire polynomial.
For example, in the polynomial \( 7p^5 - \frac{p}{10} + p^4 \), we have three terms with degrees 5, 1, and 4, respectively. The term with the highest degree is \( 7p^5 \), so the degree of the polynomial is 5.
It's important to note that coefficients (the numbers multiplying the variables) do not affect the degree of the term or the polynomial. Also, any constants (terms without variables) are considered to have a degree of 0 and are usually not included when determining the degree of a polynomial unless all other terms are also constants.
