Create a polynomial p which has the desired characteristics....

Create a polynomial $p$ which has the desired characteristics. You may leave the polynomial in factored form. $\bullet$ $p$ is degree 4 . $\bullet$ as $x \rightarrow \infty, p(x) \rightarrow-\infty$ $\bullet$ $p$ has exactly three $x$ -intercepts: (-6,0),(1,0) and (117,0) $\bullet$ The graph of $y=p(x)$ crosses through the $x$ -axis at (1,0) .

Create a polynomial $p$ which has the desired characteristics. You may leave the polynomial in factored form.
$\bullet$ $p$ is degree 4 .
$\bullet$ as $x \rightarrow \infty, p(x) \rightarrow-\infty$
$\bullet$ $p$ has exactly three $x$ -intercepts: (-6,0),(1,0) and (117,0)
$\bullet$ The graph of $y=p(x)$ crosses through the $x$ -axis at (1,0) .

Key Concepts

End Behavior of Polynomials

The end behavior of a polynomial is determined primarily by its degree and the sign of its leading coefficient. For even-degree polynomials, if the leading coefficient is positive, the polynomial tends to positive infinity in both directions, whereas if it is negative, it tends to negative infinity. This concept is critical when designing a polynomial to have a specific behavior as x approaches infinity or negative infinity.

Polynomial Construction

This concept involves building a polynomial using its zeros and their corresponding multiplicities. In constructing a polynomial in factored form, each factor (x - r) corresponds to a zero r of the polynomial. The multiplicity of a zero indicates how many times the corresponding factor is repeated in the factored form, and the product of these factors (often multiplied by a leading coefficient) yields the desired polynomial.

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero is repeated in a polynomial. When a polynomial is required to have a certain degree and a given number of distinct zeros, at least one zero might need to have a multiplicity greater than one to account for the total degree. Additionally, the multiplicity affects the graph's behavior at the intercept: if the multiplicity is odd, the graph will cross the x-axis at that zero; if even, the graph will touch the axis without crossing.

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