What does the degree of a polynomial expression tell you about its related polynomial function? Explain your thinking. Give an example of a polynomial expression of degree three. Provide information regarding the graph and zeros of the related polynomial function.
the function has the same degree as the expression.
2x^3 + 4x^2 -5x + 10
See what you can do with the graph and zeroes.
The degree of a polynomial expression indicates the highest power of the variable in the expression. In terms of the related polynomial function, the degree tells us the maximum number of roots or zeros the function can have.
For example, let's consider the polynomial expression: f(x) = 3x^3 - 2x^2 + 5x + 1
In this case, the degree of the polynomial expression is 3 because the term with the highest power of x is x^3. This means that the related polynomial function, f(x), is a cubic polynomial.
Regarding the graph of the polynomial function, since it is a cubic function with a positive leading coefficient (3 in this case), the graph will have arm-like shapes that can either rise to the left and fall to the right or fall to the left and rise to the right. The specific shape will depend on the coefficients of the other terms.
As for the zeros of the related polynomial function, a polynomial of degree three can have up to three zeros. These zeros may be real or complex numbers. To find the zeros, we can set the polynomial equal to zero and solve for x. The number of zeros can be determined by counting the number of distinct solutions.
For example, if we set the polynomial expression f(x) = 3x^3 - 2x^2 + 5x + 1 equal to zero (f(x) = 0), we can solve for x to find the zeros of the related polynomial function. The solutions we obtain will be the x-coordinates of the points where the graph crosses the x-axis.
Keep in mind that finding actual values for the zeros might involve the use of algebraic techniques such as factoring, synthetic division, or numerical methods such as Newton's method.
The degree of a polynomial expression tells you the highest power of the variable in the polynomial. It corresponds to the number of terms in the polynomial function.
For example, let's take a polynomial expression of degree three: f(x) = 2x^3 - 4x^2 + 5x - 3. In this case, the highest power of x is 3, so the degree of the polynomial is 3.
The degree of a polynomial function gives us important information about its behavior. Here's what it tells us:
1. The leading coefficient: The coefficient of the term with the highest power of the variable. In our example, the leading coefficient is 2. It helps determine the "end behavior" of the graph, meaning whether the graph goes up to positive infinity or down to negative infinity as x goes to positive or negative infinity.
2. The number of roots or zeros: The number of times the polynomial function crosses or touches the x-axis. In general, a polynomial of degree n can have at most n distinct roots.
In our example, since the degree of the polynomial is 3, there could be up to 3 zeros or roots. The graph of this polynomial function can intersect or touch the x-axis at a maximum of three points.
To find the zeros of a polynomial function, we set f(x) equal to zero (f(x) = 0) and solve for x. Zeros represent the x-values where the function equals zero.
However, keep in mind that not all polynomial functions of degree 3 will have three distinct zeros. Some may have repeated zeros, where the graph just touches the x-axis without crossing it multiple times.
To summarize, the degree of a polynomial expression provides information about the highest power of the variable, the leading coefficient, and the possible number of zeros or roots in the related polynomial function.