Finding rational zeros: x^4+2x^3+4x^2+2x+5

[koreanstar88]

for example this problem:

x^4+2x^3+4x^2+2x+5

is there an easier way to find rational zeros without using synthetic division and listing out all the possible numbers that would make it zero? because i feel like it takes some time to work out these problems using synthetic division. and i wanna know if theres a way to find them without much difficulty. please help thanks!!

 

[jwpaine]

Re: Finding rational zeros..i need help soon! thanksss

for example this problem:

x^4+2x^3+4x^2+2x+5

is there an easier way to find rational zeros without using synthetic division and listing out all the possible numbers that would make it zero? because i feel like it takes some time to work out these problems using synthetic division. and i wanna know if theres a way to find them without much difficulty. please help thanks!!

Grouping method:

let 4x^2 = [5x^2 - 1x^2]


x^4 +2x^3 +4x^2 +2x +5

= (x^4 + 2x^3 + 5x^2) - (x^2 + 2x + 5)
= x^2(x^2 + 2x + 5) - (x^2 + 2x + 5)
= (x^2 + 2x + 5)(x^2 - 1)

Does that help? Now set the factors equal to zero: complete the square for the trinomial and solve for the binomial giving you a total of four roots.

John.

 

[stapel]

On occasion, and after much practice, there will be exercises, as demonstrated in the previous reply, for which other, "faster", methods are available. In general, however, the process is as you have described: Apply the Rational Roots Test, test various possible roots with synthetic division, get yourself down to a quadratic, and apply the Quadratic Formula. Sorry!

If you find yourself taking "too long", then the best solution would be to practice. For instance, you could work out every exercise in your book for which you have the answer (or, if you have a graphing calculator, do all the exercises and then check your answers against the picture). :idea:

Have fun!

Eliz.