# Thread: Factor of the polynomial

1. ## Factor of the polynomial

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

Could someone hellp me start this question because I've no idea at all what to do. If someone could start me off i might be able to figure it out.

thanks

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

Code:
1/2]      2    -3    -11    6

1     -1    -6
------------------------------
2    -2     -12    0
the depressed polynomial is

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

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

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

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

$e^t = 3$

$e^t = -2$

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

3. Originally Posted by skeeter
Code:
1/2]      2    -3    -11    6

1     -1    -6
------------------------------
2    -2     -12    0
the depressed polynomial is

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

Hi thanks for your help. I don't really understand this part of your answer. I've never learned what a depressed polynomial is. how did you deduce the 1 -1 -6 part? thanks

4. Hi i just found a tutorial for synthetic division online. I've never even heard of it. Thanks!

5. Originally Posted by djmccabie
Hi thanks for your help. I don't really understand this part of your answer. I've never learned what a depressed polynomial is. how did you deduce the 1 -1 -6 part? thanks
Try googling synthetic division and it should make sense. I would explain to you but just not sure how to.

6. Hello, djmccabie!

a) Show that $2x-1$ is a factor of the polynomial $2x^3 - 3x^2 - 11x + 6$

Find the other factors of the polynomial
What's stopping you from using long division?

. . $\begin{array}{cccccc} & & & x^2 & -x & -6 \\
& & -- & -- & -- & -- \\ 2x-1 & ) & 2x^3 & -3x^2 & -11x & +6 \\ & & 2x^3 & -x^2 \\ & & -- & -- \\ & & & -2x^2 & -11x \\ & & & -2x^2 & +x \\ & & & -- & -- \\ & & & & -12x & +6 \\ & & & & -12x & +6 \\ & & & & -- & --\end{array}$

Therefore: . $2x^3-3x^2-11x+6 \:=\:(2x-1)(x^2-x-6) \:=\:\boxed{(2x-1)(x+2)(x-3)}$