Question 630737: show that the real zeros of each polynomial function satisfy the given conditions.
f(x)=3x^4+2x^3-4x^2+x-1; no real zero less than -2
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! I am not sure how if this the way you are supposed to solve this problem, but since no one else has responded here goes:
- f(x) has an even degree (because the highest exponent, 4, is even). This tells us that for large positive or large negative values of x, f(x) will become very large positive numbers or very large negative numbers.
- The positive coefficient in front of the
 term, 3, tells us that f(x) will become very large positive, not negative, numbers for large positive and large negative x's. On the graph, this means that f(x) will shoot up (and never some back down) for large positive x's (on the right) and for large negative x's (on the left). - From Calculus (or pre-Calculus) you might know that a 4th degree polynomial can have 3 relative maximums or minimums.
- Use a graphing calculator to graph f(x). (See below.) You can see:
- The graph shooting up on the right and left as we predicted.
- You can see that the leftmost (lowest) x-intercept is somewhere between -1 and -2.
- Three relative maximums and minimums [near (-1, -5), (0, -1) and (1/2, -2)] which is the most we can have. So there is no way for the graph to turn back downwards and hit the x-axis again somewhere out of view.
- This makes that zero between -1 and -2 the lowest one possible.
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