Each graph is of a polynomial function f(x) of degree 5...

Each graph is of a polynomial function $f(x)$ of degree 5 whose leading coefficient is $1 .$ The graph is not drawn to scale. Use the Factor Theorem to find the polynomial. [Hint: What are the roots of $f(x)$ ? What does the Factor Theorem tell you?]
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Key Concepts

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It plays a significant role in determining the polynomial's end behavior and scaling. When the leading coefficient is specified (for example, as 1), it constrains the polynomial’s form and is crucial in constructing the fully expanded polynomial after factoring.

Degree of a Polynomial

The degree of a polynomial is the highest exponent of the variable in the polynomial expression. It dictates the overall behavior of the polynomial, including the number of possible roots and the end behavior of its graph. A higher degree generally means more potential x-intercepts and more complex graph behavior.

Roots (Zeros) of a Polynomial

Roots, or zeros, of a polynomial are the solutions to the equation f(x) = 0. Each root corresponds to a value of x where the polynomial equals zero, and these values indicate the x-intercepts of the graph of the polynomial. Identifying the roots allows one to express the polynomial in its factored form using the Factor Theorem.

Polynomial Function

A polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents. Understanding the structure of a polynomial is crucial, especially when determining its factors, graphing its behavior, or solving for its roots.

Factor Theorem

The Factor Theorem asserts that a polynomial f(x) has a factor (x - r) if and only if f(r) = 0. This theorem provides a direct relationship between the roots (or zeros) of the polynomial and its factors, which is essential in constructing or factoring a polynomial when its roots are known.

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