Question

the polynomial(2x+1)(x-2)(3x-4) is greater than zero for x in (-infinity -1/2) U (4/3, 2)/
true or false?

Answer 1

f(x) = (2x+1)(x-2)(3x-4) has the roots x = -1/2, 2 and 4/3
Pick a convenient point in the interval (-infinity -1/2) and evaluate f(x).
A convenient point is a small round number such as -1 in the interval (-infinity -1/2).
Find f(-1)

Answer 2

Recall interval notation \(x\in (-\infty , -1/2)\) means \(\infty\lt x \lt -1/2\).
and simililarly for the other interval. The sign \(\cup\) means the union of the two intervals.

Sketch the function and the answer will come out easily.
Since IF we expanded it, the leading term is \(+6x^3\) meaning that the graph will go from (-,-) to (+,+), i.e. lower left to upper right.
We also know that the zeroes are \(\{-\frac{1}{2},\frac{4}{3},2\}\), we can draw the graph as follows:

Answer 3

Can you deduce the result?

Answer 4

yes thank you :)

Answer 5

You're welcome! :)