Question

1. $p(x)=-3x^{3}+bx^{2}+cx + d$ is an integral polynomial function with zeroes 2, -1, and 4. a sketch of $y = p(x)$ is shown below. at which point does the graph cross the $y$-axis? a. $(0,-8)$ b. $(0,-15)$ c. $(0,-16)$ d. $(0,-24)$ 8. the graph of a fourth - degree polynomial function of the form $p(x)=ax^{4}+bx^{3}+cx^{2}+dx + e$ is shown. the values of $a$ and $e$ must satisfy a. $a>0,e<0$ b. $a<0,e<0$ c. $a>0,e>0$ d. $a<0,e>0$ 9. numerical response - the remainder when $p(x)=2x^{4}-3x^{3}+1$ is divided by $(x - 5)$ is ______.

Explanation:

7.

Step1: Write the polynomial in factored form

Since the zero - es of \(P(x)=-3x^{3}+bx^{2}+cx + d\) are \(x = 2\), \(x=-1\), and \(x = 4\), we can write \(P(x)=-3(x - 2)(x + 1)(x - 4)\).

Step2: Expand the polynomial

\[

$$\begin{align*} P(x)&=-3(x - 2)(x^{2}-3x - 4)\\ &=-3(x^{3}-3x^{2}-4x-2x^{2}+6x + 8)\\ &=-3(x^{3}-5x^{2}+2x + 8)\\ &=-3x^{3}+15x^{2}-6x-24 \end{align*}$$

\]

Step3: Find the \(y\) - intercept

The \(y\) - intercept occurs when \(x = 0\). Substitute \(x = 0\) into \(P(x)\): \(P(0)=-24\).

Step1: Analyze the end - behavior for the leading coefficient \(a\)

As \(x\to\pm\infty\), for a fourth - degree polynomial \(y = ax^{4}+bx^{3}+cx^{2}+dx + e\), the end - behavior is determined by the leading term \(ax^{4}\). Since the graph of the polynomial rises to the left and rises to the right, \(a>0\).

Step2: Analyze the \(y\) - intercept for \(e\)

The \(y\) - intercept of the polynomial \(y = ax^{4}+bx^{3}+cx^{2}+dx + e\) is at the point \((0,e)\). From the graph, the \(y\) - intercept is below the \(x\) - axis, so \(e<0\).

Step1: Use the Remainder Theorem

According to the Remainder Theorem, if a polynomial \(P(x)\) is divided by \((x - k)\), the remainder is \(P(k)\). Here, \(P(x)=2x^{4}-3x^{3}+1\) and \(k = 5\).

Step2: Calculate \(P(5)\)

\[

$$\begin{align*} P(5)&=2(5)^{4}-3(5)^{3}+1\\ &=2\times625-3\times125 + 1\\ &=1250-375+1\\ &=875 + 1\\ &=876 \end{align*}$$

\]

Answer:

D. \((0,-24)\)

8.