Can you explain the solution for 5x3 + 2x2 + x = 20 between x = 1.4 and x = 1.5?

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I've just seen the following question on a past paper:

"Explain why 5x3 + 2x2 + x = 20 has a solution between x = 1.4 and x = 1.5. Show all your calculations."

In order to answer this question, is it sufficient to simply work out the equation twice gving x the 2 values stated in the question?
I have done that and the 2 solution are:
19.04 (if x = 1.4)
22.875 (if x = 1.5)
 
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Rarely do they explicitly teach the "intermediate value theorem" in lower algebra classes, but that's what's going on here.
You have a polynomial whose value is greater than 20 at some point (call it "b") and less than 20 at another point ("a"). The theorem states that the polynomial will "hit" every value in between f(b) = 22.875 and f(a) = 19.04 at least once.

This relies on the fact that polynomials are continuous and differentiable, but that might be a little above the scope of your math class.
 
Gringo123 said:
I've just seen the following question on a past paper:

"Explain why 5x3 + 2x2 + x = 20 has a solution between x = 1.4 and x = 1.5. Show all your calculations."

In order to answer this question, is it sufficient to simply work out the equation twice gving x the 2 values stated in the question?
I have done that and the 2 solution are:
19.04 (if x = 1.4)
22.875 (if x = 1.5)

Yes, this is sufficient. Here's a highly intuitive (read that as non-mathematically based) explanation why.

think about the graph of your polynomial. If you were to draw it, or have a computer draw it, it would appear in one piece - no gaps, no places where there are tears or holes. What you've shown is that at x = 1.4, the height of the graph is just smaller than 20, and at x = 1.5, the height of the graph is just higher than 20. Since there aren't any holes, somewhere between 1.4 and 1.5 there has to be a place where the graph's height is exactly 20.

this is the point of The Chaz's post.
 
You have 888 posts! Nice. I'm planning to quit at (exactly) that number!
Also, props for using "The" (Chaz). It's an oft-forgotten article of much importance ;)