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**djmccabie** a) Show that 2x-1 is a factor of the polynomial 2x³ - 3x² - 11x + 6

Find the other factors of the polynomial

b) Find, correct to three decimal places, the values of t satisfying

2e^3t - 3e^2t - 11e^t + 6 = 0

a) if $\displaystyle 2x-1$ is a factor, then $\displaystyle f\left(\frac{1}{2}\right) = 0$ ... show this by synthetic division

Code:

1/2] 2 -3 -11 6
1 -1 -6
------------------------------
2 -2 -12 0

the depressed polynomial is

$\displaystyle 2x^2 - 2x - 12 = 2(x^2 - x - 6) = 2(x - 3)(x + 2)$

the other two roots are $\displaystyle x = 3$ and $\displaystyle x = -2$

b) note the similarity between this equation and the one in part (a)

$\displaystyle e^t = \frac{1}{2}$

$\displaystyle e^t = 3$

$\displaystyle e^t = -2$

two of the three above equations will yield a real solution.