In the context of polynomials, the degree of a term is the sum of the exponents of the variables in that term. The degree of a polynomial is the highest degree of any of its terms.
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The degree of a constant term (a term with no variables) is always 0.
The degree of a variable raised to a power is the value of that power.
The degree of a product of terms is the sum of the degrees of the individual terms.
The degree of a sum or difference of polynomials is the highest degree of any of the individual terms.
Knowing the degree of a polynomial is important for understanding the behavior and properties of the polynomial, such as its end behavior and the number of real roots it may have.
Review Questions
How do you determine the degree of a polynomial?
To determine the degree of a polynomial, you need to find the highest exponent of the variable(s) in the polynomial. The degree of a term is the sum of the exponents of the variables in that term, and the degree of the polynomial is the highest degree of any of its terms. For example, in the polynomial $5x^3 - 2x^2 + 4x - 1$, the degrees of the terms are 3, 2, 1, and 0, respectively, so the degree of the entire polynomial is 3.
Explain the relationship between the degree of a polynomial and the number of real roots it may have.
The degree of a polynomial is related to the number of real roots it may have. Specifically, a polynomial of degree $n$ can have at most $n$ distinct real roots. This means that a linear polynomial (degree 1) can have at most 1 real root, a quadratic polynomial (degree 2) can have at most 2 real roots, a cubic polynomial (degree 3) can have at most 3 real roots, and so on. However, it's important to note that a polynomial may have fewer real roots than its degree, and it may also have complex roots in addition to real roots.
How does the degree of a polynomial affect its end behavior?
The degree of a polynomial has a significant impact on its end behavior, which refers to the behavior of the polynomial as the input variable approaches positive or negative infinity. Specifically, the sign of the leading coefficient (the coefficient of the term with the highest degree) and the degree of the polynomial determine whether the polynomial will approach positive or negative infinity as the input variable increases or decreases. For example, a polynomial of odd degree will have an end behavior that approaches positive or negative infinity, depending on the sign of the leading coefficient, while a polynomial of even degree will approach positive infinity regardless of the sign of the leading coefficient.