The degree of a polynomial is the highest exponent of the variable(s) in the polynomial expression. It determines the complexity and behavior of the polynomial function.
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The degree of a polynomial is the highest exponent of the variable(s) in the expression.
The degree of a monomial is simply the exponent of the variable.
The degree of a binomial or a polynomial with more than two terms is the sum of the exponents of the variables in the term with the highest total exponent.
The degree of a constant polynomial (a polynomial with no variables) is 0.
The degree of a polynomial is an important characteristic that affects the behavior and properties of the function.
Review Questions
How do you determine the degree of a polynomial expression?
To determine the degree of a polynomial expression, you need to identify the term with the highest exponent of the variable(s). The degree of the polynomial is the sum of the exponents in that term. For example, in the polynomial $4x^3 + 2x^2 - 5x + 7$, the term with the highest exponent is $4x^3$, so the degree of the polynomial is 3.
Explain how the degree of a polynomial affects its behavior and properties.
The degree of a polynomial is a crucial characteristic that determines its behavior and properties. A polynomial of degree $n$ will have at most $n$ real roots, and its graph will have at most $n-1$ turning points. The degree also affects the end behavior of the function, with higher degree polynomials tending to grow or decay more rapidly as the input values increase or decrease. Understanding the degree of a polynomial is essential for analyzing its features, such as its critical points, local extrema, and asymptotic behavior.
How does the degree of a polynomial relate to the process of multiplying polynomials, as discussed in section 10.3?
When multiplying two polynomials, the degree of the resulting polynomial is the sum of the degrees of the original polynomials. This is because the exponents of the variables in each term are added together during the multiplication process. For example, if you multiply a polynomial of degree $m$ by a polynomial of degree $n$, the resulting polynomial will have a degree of $m + n$. This relationship between the degrees of the factors and the degree of the product is a crucial concept in the context of multiplying polynomials, as discussed in section 10.3.