SOLUTION: Use the intermediate value theorem to show that f(x) has a zero in the given interval. Please show all of your work. f(x)= -3x^4-5x^3-8x+7; [-2.4,-2.3]

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Question 530888: Use the intermediate value theorem to show that f(x) has a zero in the given interval. Please show all of your work.
f(x)= -3x^4-5x^3-8x+7; [-2.4,-2.3]
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!
Since f(x) is a polynomial, we only have to show that 
f(-2.4) and f(-2.3) have different signs. 

There are two ways to find f(-2.4) and f(-2.3)

Method I.  Direct substitution:

f(-2.4) = -3(-2.4)* - 5(-2.4)³ - 8x + 7 
        = -3(33.1776) - 5(-13.824) - 8(-2.4) + 7
        = -99.5328 + 69.12 + 19.2 + 7
        = -4.2128 

f(-2.3) = -3(-2.3)* - 5(-2.3)³ - 8x + 7 
        = -3(27.9841) - 5(-12.9841) - 8(-2.3) + 7
        = -83.9523 + 60.835 + 18.4 + 7
        =  2.2827

They have different signs, so that's all you need to guarantee
that a polynomial has a zero in the interval between the two
values of x. 

Method II.  The remainder theorem (synthetic division)

-2.4 | -3  -5        0  -8        7
     |      7.2  -5.28  12.672  -11.2128
       -3   2.2  -5.28   4.672   -4.2128

So f(-2.4) = the remainder = -4.2128

-2.3 | -3  -5        0  -8       7
     |      6.9  -4.37  10.051  -4.7173
       -3   1.9  -4.37   2.051   2.2827

So f(-2.3) = the remainder = 2.2827

They have different signs, so that's all you need to guarantee
that a polynomial has a zero in the interval between the two
values of x.

Edwin
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