The factor of the polynomial $6 x^{4} + 29 x^{3} + 30 x^{2} - 11 x - 6$

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x + 3

x + 2

x - 2

2 x - 1

answer is A, B.

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Detailed Solution

f (x) = 6 x^{4} + 29 x^{3} + 30 x^{2} - 11 x - 6

$f (- 2) = 6 {(- 2)}^{4} + 29 {(- 2)}^{3} + 30 {(- 2)}^{2} - 11 (- 2) - 6$

$= 96 - 232 + 120 + 22 - 6$

$= - 238 + 238$

$= 0$

$∴ (x + 2)$ is a factor of $6 x^{4} + 29 x^{3} + 30 x^{2} - 11 x - 6$

$f (\frac{1}{2}) = 6 \times \frac{1}{16} + 29 \times \frac{1}{8} + 30 \times \frac{1}{4} - 11 \times \frac{1}{2} - 6$

$= \frac{6}{16} + \frac{29}{8} + \frac{30}{4} - \frac{11}{2} - 6$

$= \frac{6 + 58 + 120 - 88 - 96}{16}$

$= \frac{184 - 184}{16}$

$= 0$

$∴ (2 x - 1)$ is a factor of $6 x^{4} + 29 x^{3} + 30 x^{2} - 11 x - 6$

$f (2) = 6 {(2)}^{4} + 29 {(2)}^{3} + 30 {(2)}^{2} - 11 (2) - 6$

$= 96 + 232 + 120 - 22 - 6$

$= 420$

$(x - 2)$ is not a factor of $6 x^{4} + 29 x^{3} + 30 x^{2} - 11 x - 6$

$f (- 3) = 6 {(- 3)}^{4} + 29 {(- 3)}^{3} + 30 (- 3) - 11 (- 3) - 6$

$= 486 - 783 - 90 + 33 - 6$

$= 519 - 873$

$= - 354$

$(x + 3)$ is not a factor of $6 x^{4} + 29 x^{3} + 30 x^{2} - 11 x - 6$

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