Question

Find the zeros of g(x)=4x3+13x2−37x−10, given its graph. (1 point) Responses x={−10,−5,−14,2} x equals left brace negative 10 comma negative 5 comma negative Start Fraction 1 over 4 End Fraction comma 2 right brace x={−5,−14,2} x equals left brace negative 5 comma negative Start Fraction 1 over 4 End Fraction comma 2 right brace x={−5,2} x equals left brace negative 5 comma 2 right brace x={−10,−5,2}

Answer 1

To find the zeros of the polynomial function \( g(x) = 4x^3 + 13x^2 - 37x - 10 \), we usually look for values of \( x \) for which \( g(x) = 0 \).

Since this problem suggests that there's a graph involved, let's evaluate the potential zero candidates from the options provided:

1. \( x = -10 \)
2. \( x = -5 \)
3. \( x = -\frac{1}{4} \)
4. \( x = 2 \)

To find the zeros, we can use synthetic division or direct substitution to see which of these values results in \( g(x) = 0 \).

### Checking the Possible Zeros:
1. **For \( x = -10 \)**:
\[
g(-10) = 4(-10)^3 + 13(-10)^2 - 37(-10) - 10 \\
= 4(-1000) + 13(100) + 370 - 10 \\
= -4000 + 1300 + 370 - 10 \\
= -4000 + 1660 = -2340 \quad (\text{not a zero})
\]

2. **For \( x = -5 \)**:
\[
g(-5) = 4(-5)^3 + 13(-5)^2 - 37(-5) - 10 \\
= 4(-125) + 13(25) + 185 - 10 \\
= -500 + 325 + 185 - 10 \\
= -500 + 500 = 0 \quad (\text{this is a zero})
\]

3. **For \( x = -\frac{1}{4} \)**:
\[
g\left(-\frac{1}{4}\right) = 4\left(-\frac{1}{4}\right)^3 + 13\left(-\frac{1}{4}\right)^2 - 37\left(-\frac{1}{4}\right) - 10 \\
= 4\left(-\frac{1}{64}\right) + 13\left(\frac{1}{16}\right) + \frac{37}{4} - 10 \\
= -\frac{1}{16} + \frac{13}{16} + \frac{37}{4} - 10 \\
= \frac{12}{16} + \frac{37}{4} - 10 \\
= \frac{3}{4} + \frac{37}{4} - 10 = \frac{40}{4} - 10 = 10 - 10 = 0 \quad (\text{this is also a zero})
\]

4. **For \( x = 2 \)**:
\[
g(2) = 4(2)^3 + 13(2)^2 - 37(2) - 10 \\
= 4(8) + 13(4) - 74 - 10 \\
= 32 + 52 - 74 - 10 \\
= 84 - 84 = 0 \quad (\text{this is a zero})
\]

### Conclusion:
The zeros of the polynomial \( g(x) = 4x^3 + 13x^2 - 37x - 10 \) are:
\[
x = -5, \, -\frac{1}{4}, \, 2
\]

From the provided options, the correct response is:
\[
\boxed{x = \{-5, -\frac{1}{4}, 2\}}
\]