Write a polynomial function whose graph is shown (use the smallest degree possible).

Name: Problem 80, Chapter 5: College Algebra
Uploaded: 2020-05-13T16:28:46-08:00
Duration: 1 min 46 s
Channel: Ankit Gupta
Description: Write a polynomial function whose graph is shown (use the smallest degree possible).

step 1 from the graph what we can conclude is that it is representing a polynomial f of x with its zero

Write a polynomial function whose graph is shown (use the...

Key Concepts

Degree of the Polynomial

The degree of the polynomial is the highest power of x in the expression and dictates the maximum number of turning points the graph can have. When crafting a polynomial from a graph, using the smallest possible degree means incorporating only the necessary factors and using the minimal multiplicities required to match the graph's behavior.

Factorization and the Factor Theorem

Using the Factor Theorem, each zero of the polynomial corresponds to a linear factor of the form (x - r), where r is a root. Constructing the polynomial in its factored form by multiplying these factors together (and including an appropriate leading coefficient) is a standard method to recreate the graph of the polynomial with the smallest possible degree.

Polynomial Functions

A polynomial function is an expression involving a sum of powers in one or more variables multiplied by coefficients. In constructing a polynomial from a graph, you identify the roots, the highest degree term, and how the function behaves at the extremes, ensuring that the expression accurately captures the shape of the graphed curve.

Zeros and Multiplicities

The zeros (or roots) of the polynomial are the x-values where the function equals zero, corresponding to the points where the graph crosses or touches the x-axis. The multiplicity of each zero affects the behavior of the graph at that point—if a zero has an odd multiplicity, the graph crosses the axis, and if it has an even multiplicity, the graph only touches the axis and turns around.

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