I am trying to factor a 4th degree polynomial that does not have any rational roots. I need to somehow get it factored into two quadratics. Anyone know of a method to use.
3x^4 - 8x^3 - 5x^2 + 16x - 5
Two of the irrational roots are 1.135.. and 0.382.. but that won't help you with the factoring.
You might try factoring the fourth order polynomial into two quadratics of the form
(3x^2 + ax + b)(x^2 + cx + d)
and picking a, b, c and d to get the coefficents in the fourth order polynomial to agree. For example,
bd = -5
3c + a = -8
ad + bc = 16
3d + b + ac = -5
Where did you get that form of factoring? I have never seen any books that address factoring a fourth degree polynomial like the one I presented.
Thank you very much.
I got it from your suggestion that it be factored into quadratics. I got the two real roots by essentially graphical means. There is a specific formula for the roots of fourth order polynomials, but it is very long and complicated. I think they are hoping you will find it by trial and error.
I could understand getting the roots by graphical means if the book had put the answers in decimal form. However, they had the answers in exact form and I knew the only way to get them would be by solving a quadratic equation. Thanks once again for your help.
You're welcome! I can help explain the process of factoring a 4th degree polynomial as well as the method you mentioned using two quadratics.
Factoring a 4th degree polynomial can be challenging, especially when it doesn't have any rational roots. In such cases, one approach is to try factoring it into two quadratics. Here's how you can do it:
1. Write the polynomial in the form:
f(x) = a*x^4 + b*x^3 + c*x^2 + d*x + e.
2. Assume that the polynomial can be factored into two quadratics in the form:
f(x) = (p*x^2 + q*x + r)(s*x^2 + t*x + u).
3. Multiply out the two quadratics and equate the corresponding coefficients with the original polynomial:
p*s = a (coefficient of x^4 term)
p*t + q*s = b (coefficient of x^3 term)
p*u + q*t + r*s = c (coefficient of x^2 term)
q*u + r*t = d (coefficient of x term)
r*u = e (constant term)
4. Solve this system of equations to find the values of p, q, r, s, t, and u.
In your case, the polynomial is given as 3x^4 - 8x^3 - 5x^2 + 16x - 5. Let's apply the method:
Assume the polynomial can be factored into two quadratics:
(3x^2 + ax + b)(x^2 + cx + d).
Equating the coefficients, we have:
b * d = -5
3c + a = -8
a * d + b * c = 16
3d + b + a * c = -5
Solve this system of equations to find the values of a, b, c, and d. You mentioned that two irrational roots are approximately 1.135 and 0.382, which can be helpful in the process.
Please note that factoring higher-degree polynomials can sometimes be challenging and may not always be possible using simple methods. In such cases, numerical methods or software tools may be required.
I hope this explanation clarifies the method you mentioned. Let me know if you need any further assistance!