Bot GPT 4-o mini To determine whether \(3x + 2\) is a factor of \(6x^4 - 11x^3 - 7x^2 + 8x + 3\), we can use polynomial long division or synthetic division. However, a quicker method is to check if the polynomial evaluates to zero when the root of \(3x + 2\) is substituted.
First, let's find the root of \(3x + 2 = 0\):
\[ 3x + 2 = 0 \implies 3x = -2 \implies x = -\frac{2}{3} \]
Now we will substitute \(x = -\frac{2}{3}\) into the polynomial \(6x^4 - 11x^3 - 7x^2 + 8x + 3\):
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Calculate \(6\left(-\frac{2}{3}\right)^4\): \[ 6\left(-\frac{2}{3}\right)^4 = 6 \times \frac{16}{81} = \frac{96}{81} \]
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Calculate \(-11\left(-\frac{2}{3}\right)^3\): \[ -11\left(-\frac{2}{3}\right)^3 = -11 \times -\frac{8}{27} = \frac{88}{27} \]
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Calculate \(-7\left(-\frac{2}{3}\right)^2\): \[ -7\left(-\frac{2}{3}\right)^2 = -7 \times \frac{4}{9} = -\frac{28}{9} \]
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Calculate \(8\left(-\frac{2}{3}\right)\): \[ 8\left(-\frac{2}{3}\right) = -\frac{16}{3} \]
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The constant term is \(3\) which can be written as \(\frac{243}{81}\) for common denominator purposes.
Summing these values together: \[ \frac{96}{81} + \frac{88}{27} - \frac{28}{9} - \frac{16}{3} + 3 \] We convert everything to the common denominator of \(81\): \[ \frac{96}{81} + \frac{264}{81} - \frac{252}{81} - \frac{432}{81} + \frac{243}{81} \]
Now summing these: \[ \frac{96 + 264 - 252 - 432 + 243}{81} = \frac{-81}{81} = -1 \]
Since the result is \(-1\), it is not equal to zero. Therefore, \(3x + 2\) is not a factor of the polynomial.
The answer is:
B. No
1 answer