Show that each polynomial function has a real zero as described i...

Question

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=6x^4+13x^3-11x^2-3x+5no zero less than -3

Accepted Answer

For the following polynomial function prove that it has no zero, less than negative two where we have nine X to the fourth plus five X to the third minus six X square minus 15 X plus eight. Now to check if it has no zeros, less than negative two, bill will make use of the badness theorem which states if we're dividing our function by a factor X minus C, if that C is less than zero and the resulting sedate division has alternating signs, then F FX has no zero less than C. Now we noticed we do have a negative C, our C is negative two. So let's do a synthetic division to see if we get alternating terms. So we have negative two and we have our polynomial nine, five, negative six, negative 15 and eight with synthetic division, we will first bring down our nine, then multiply our resulting non by our negative two. That is a of 18, we can then add these together five plus negative 18, gives us negative 13. Then we'll repeat negative 13, multiplied by negative two. Gives us 26 native six plus 26 gives us 20 and neither two will give us neither 40 native 15 minus 40 is native 55. Finally negative 55 multiplied by negative two, just 1 10, eight plus 1 10 will be 1 18, which is our remainder. Now, we want to check if we have alternating signs in our result. We notice here we go from nine to negative 13, negative 13 to 20 and 20 to negative 55. So we have multiple places where the signs change because that is the case, we will confirm by answer B this yield an expression. There were terms in order of decreasing degree which have alternating signs. So we have proved that there is no zero, less than negative two. OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

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Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=6x^4+13x^3-11x^2-3x+5no zero less than -3

Zeros of Polynomial Functions practice set