rational roots theorem

[slockheart]

let a1, a2, and a3 be integers with prime number c as an integral factor.

a. prove and integral root of x^3 + a1x^2 + a2x + a3 = 0 musrt have c as an integral factor

b. use this result to find rational roots of x^4 + 24x^3 - 52x^2 +26x - 52 = 0

i think im supposed to be using the rational roots theorem to do this.

 

[stapel]

let a1, a2, and a3 be integers with prime number c as an integral factor.

a. prove and integral root of x^3 + a1x^2 + a2x + a3 = 0 musrt have c as an integral factor

Use the fact that "c" is a factor of a[sub:1xgcqomn]1[/sub:1xgcqomn], a[sub:1xgcqomn]2[/sub:1xgcqomn], and a3[/sup] to rewrite these three coefficients in terms of "c". Then apply the Rational Roots Test that you learned about back in algebra.

b. use this result to find rational roots of x^4 + 24x^3 - 52x^2 +26x - 52 = 0

Find a common factor of 24, 52, and 26. :wink:

Eliz.

 

[Pklarreich]

Suppose R is an integral root. Then

P(R) = 0 ==>
R^3 + a1 R^2 + a2 R + a3 = 0.

Now if c divides a1,a2,a3, it divides (a1 R^2 + a2 R + a3).
Therefore it divides R^3. And a prime that divides a product must divide one of the factors.

You can take it from there.