Question

Use the remainder theorem and synthetic division to evaluate f(t) =2t^5-5t^4-8t+20 at (-4). Write as a product of linear factors and determine all roots

Answer 1

This does not look right to me this is what I got after the synthetic division:
2t^4-13t^2+52t^2-208t+824 remainder -3276. Just doesn't seem right

Answer 2

Actually, it is correct! Try grinding out \[2(-4)^5-5(-4)^4-8(-4)+20\] and see what you get. Synthetic substitution sure beats doing it this way, doesn't it?

Answer 3

No way, I actually did it correctly, i'll see what I get!

Answer 4

I get confused when it comes to the product of linear factors. Is what I came up with the product of linear factors, or do I need to do something more and I don't think there are any roots because there is the remainder correct?

Answer 5

Linear factor just means a polynomial with no exponents other than 1. \((x-3)\) is a linear factor, \((x^2-3)\) is not.

Any polynomial you can factor can be written as a product of factors. For each factor, there will be a root which makes that factor equal to 0, and the zero product principle lets us figure out the roots by factoring, setting each factor equal to 0, and solving for the value of \(x\) that makes it so. If your polynomial can be factored into nothing but linear factors, you can decompose it by successively using synthetic division (which only works with linear factors) until there's nothing left but 1. Of course, you'll need some idea of what to use as divisors, and there the rational root theorem can give you assistance, even if sometimes it gives you a bit too much "assistance" :-)

Did I succeed in confusing you more, or do I need to try again? :-)