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:
- \( x = -10 \)
- \( x = -5 \)
- \( x = -\frac{1}{4} \)
- \( x = 2 \)
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:
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For \( x = -10 \): \[ g(-10) = 4(-10)^3 + 13(-10)^2 - 37(-10) - 10 \ = 4(-1000) + 13(100) + 370 - 10 \ = -4000 + 1300 + 370 - 10 \ = -4000 + 1660 = -2340 \quad (\text{not a zero}) \]
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For \( x = -5 \): \[ g(-5) = 4(-5)^3 + 13(-5)^2 - 37(-5) - 10 \ = 4(-125) + 13(25) + 185 - 10 \ = -500 + 325 + 185 - 10 \ = -500 + 500 = 0 \quad (\text{this is a zero}) \]
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For \( x = -\frac{1}{4} \): \[ g\left(-\frac{1}{4}\right) = 4\left(-\frac{1}{4}\right)^3 + 13\left(-\frac{1}{4}\right)^2 - 37\left(-\frac{1}{4}\right) - 10 \ = 4\left(-\frac{1}{64}\right) + 13\left(\frac{1}{16}\right) + \frac{37}{4} - 10 \ = -\frac{1}{16} + \frac{13}{16} + \frac{37}{4} - 10 \ = \frac{12}{16} + \frac{37}{4} - 10 \ = \frac{3}{4} + \frac{37}{4} - 10 = \frac{40}{4} - 10 = 10 - 10 = 0 \quad (\text{this is also a zero}) \]
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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: \[ \boxed{x = {-5, -\frac{1}{4}, 2}} \]
3 answers