Question

• 4x³ + 9x² + 2x - 5 to complete these sentences.

= 2 is ______.

1. state all the intervals in which the zeros are located.

f(x) = x⁴ + 4x³ - 9x² - 37x - 24

Explanation:

Step1: Evaluate \( f(-5) \)

Substitute \( x = -5 \) into \( f(x) = x^4 + 4x^3 - 9x^2 - 37x - 24 \):
\[

$$\begin{align*} f(-5) &= (-5)^4 + 4(-5)^3 - 9(-5)^2 - 37(-5) - 24 \\ &= 625 + 4(-125) - 9(25) + 185 - 24 \\ &= 625 - 500 - 225 + 185 - 24 \\ &= (625 - 500) + (-225 + 185) - 24 \\ &= 125 - 40 - 24 \\ &= 61 \end{align*}$$

\]

Step2: Evaluate \( f(-4) \)

Substitute \( x = -4 \) into \( f(x) \):
\[

$$\begin{align*} f(-4) &= (-4)^4 + 4(-4)^3 - 9(-4)^2 - 37(-4) - 24 \\ &= 256 + 4(-64) - 9(16) + 148 - 24 \\ &= 256 - 256 - 144 + 148 - 24 \\ &= (256 - 256) + (-144 + 148) - 24 \\ &= 0 + 4 - 24 \\ &= -20 \end{align*}$$

\]

Step3: Evaluate \( f(-1) \)

Substitute \( x = -1 \) into \( f(x) \):
\[

$$\begin{align*} f(-1) &= (-1)^4 + 4(-1)^3 - 9(-1)^2 - 37(-1) - 24 \\ &= 1 + 4(-1) - 9(1) + 37 - 24 \\ &= 1 - 4 - 9 + 37 - 24 \\ &= (1 - 4) + (-9 + 37) - 24 \\ &= -3 + 28 - 24 \\ &= 1 \end{align*}$$

\]

Step4: Evaluate \( f(0) \)

Substitute \( x = 0 \) into \( f(x) \):
\[
f(0) = 0^4 + 4(0)^3 - 9(0)^2 - 37(0) - 24 = -24
\]

Step5: Evaluate \( f(3) \)

Substitute \( x = 3 \) into \( f(x) \):
\[

$$\begin{align*} f(3) &= 3^4 + 4(3)^3 - 9(3)^2 - 37(3) - 24 \\ &= 81 + 4(27) - 9(9) - 111 - 24 \\ &= 81 + 108 - 81 - 111 - 24 \\ &= (81 + 108) + (-81 - 111) - 24 \\ &= 189 - 192 - 24 \\ &= -27 \end{align*}$$

\]
Wait, that can't be right. Wait, let's recalculate \( f(3) \):
\[

$$\begin{align*} f(3) &= 81 + 4\times27 - 9\times9 - 37\times3 - 24 \\ &= 81 + 108 - 81 - 111 - 24 \\ &= (81 - 81) + (108 - 111) - 24 \\ &= 0 - 3 - 24 \\ &= -27 \end{align*}$$

\]
Wait, maybe I made[LLM SSE On Failure]

Answer:

Step1: Evaluate \( f(-5) \)

Substitute \( x = -5 \) into \( f(x) = x^4 + 4x^3 - 9x^2 - 37x - 24 \):
\[

$$\begin{align*} f(-5) &= (-5)^4 + 4(-5)^3 - 9(-5)^2 - 37(-5) - 24 \\ &= 625 + 4(-125) - 9(25) + 185 - 24 \\ &= 625 - 500 - 225 + 185 - 24 \\ &= (625 - 500) + (-225 + 185) - 24 \\ &= 125 - 40 - 24 \\ &= 61 \end{align*}$$

\]

Step2: Evaluate \( f(-4) \)

Substitute \( x = -4 \) into \( f(x) \):
\[

$$\begin{align*} f(-4) &= (-4)^4 + 4(-4)^3 - 9(-4)^2 - 37(-4) - 24 \\ &= 256 + 4(-64) - 9(16) + 148 - 24 \\ &= 256 - 256 - 144 + 148 - 24 \\ &= (256 - 256) + (-144 + 148) - 24 \\ &= 0 + 4 - 24 \\ &= -20 \end{align*}$$

\]

Step3: Evaluate \( f(-1) \)

Substitute \( x = -1 \) into \( f(x) \):
\[

$$\begin{align*} f(-1) &= (-1)^4 + 4(-1)^3 - 9(-1)^2 - 37(-1) - 24 \\ &= 1 + 4(-1) - 9(1) + 37 - 24 \\ &= 1 - 4 - 9 + 37 - 24 \\ &= (1 - 4) + (-9 + 37) - 24 \\ &= -3 + 28 - 24 \\ &= 1 \end{align*}$$

\]

Step4: Evaluate \( f(0) \)

Substitute \( x = 0 \) into \( f(x) \):
\[
f(0) = 0^4 + 4(0)^3 - 9(0)^2 - 37(0) - 24 = -24
\]

Step5: Evaluate \( f(3) \)

Substitute \( x = 3 \) into \( f(x) \):
\[

$$\begin{align*} f(3) &= 3^4 + 4(3)^3 - 9(3)^2 - 37(3) - 24 \\ &= 81 + 4(27) - 9(9) - 111 - 24 \\ &= 81 + 108 - 81 - 111 - 24 \\ &= (81 + 108) + (-81 - 111) - 24 \\ &= 189 - 192 - 24 \\ &= -27 \end{align*}$$

\]
Wait, that can't be right. Wait, let's recalculate \( f(3) \):
\[

$$\begin{align*} f(3) &= 81 + 4\times27 - 9\times9 - 37\times3 - 24 \\ &= 81 + 108 - 81 - 111 - 24 \\ &= (81 - 81) + (108 - 111) - 24 \\ &= 0 - 3 - 24 \\ &= -27 \end{align*}$$

\]
Wait, maybe I made[LLM SSE On Failure]