Use the Rational Zeros Theorem to find all the real zeros
Favorites|Homepage
Subscriptions | sitemap
HOME > > Use the Rational Zeros Theorem to find all the real zeros

Use the Rational Zeros Theorem to find all the real zeros

[From: ] [author: ] [Date: 12-04-26] [Hit: ]
Use the zeros to factor f over the real numbers.I dont know how to do this and I would like to know the steps to figure it out,1,thank you!-The rational roots theorem says that IF there is a rational root of a polynomial f with interger coefficients, and a/b is such a root,......
Use the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers.

f(x)=3x^4 - 7x^3 + 13x^2 - 21x + 12

I don't know how to do this and I would like to know the steps to figure it out, I do know the answer is
1, 3; f(x)= (x-1)(3x-4)(x^2+3)

thank you!

-
The rational roots theorem says that IF there is a rational root of a polynomial f with interger coefficients, and a/b is such a root, in lowest terms (a and b have no common factors), then a divides the constant term, and b divides the coefficient of the largest power of x.

So any rational root of f = a/b would require that a divide 12 (meaning a could only be 1, 2, 3, 4, 6 or 12), and b would have to divide 3, meaning it could only be 1 or 3. Oh yeah, it can be positive or negative too. So technically I should have included -1 -2 -3 etc. in the possibilities for a.

From there, you have to start testing the possibilities by picking a and b from the list of possibilities, and plugging them in. It's usually easiest to start with the integers, especially since 1 turns out to be a root.

That can be a lot of work, graphing the function can help focus the search. Eventually you would find that 4/3 is a root, meaning (x - 4/3) divides f. You'd multiply by 3 to get integer coefficients, which is where (3x-4) comes from. After you divide f by (x-1) and (4x-3), you're left with x^2+3 (I assume your answer is correct--I didn't check it), which has no real roots -- obviously the other 2 roots are +/- √3.
1
keywords: find,all,Rational,zeros,Use,real,Theorem,to,Zeros,the,Use the Rational Zeros Theorem to find all the real zeros
New
Hot
© 2008-2010 http://www.science-mathematics.com . Program by zplan cms. Theme by wukong .