Factoring Polynomial z^4-4z^3+6z^2-4z-15 =0

In summary, the conversation is about factoring a polynomial and finding its solutions using Ruffini's rule. The polynomial is divided into two factors, one of which is quadratic and the other involves finding another root. The solutions are -1, 3, and 1+/-2i. The conversation also discusses the importance of knowing the solutions in order to properly factor the polynomial.
  • #1
Fabio010
85
0
z^4-4z^3+6z^2-4z-15 =0

How can i factor this polynomial in order to find the solutions??


I tried with the ruffini' rule.

and i reached the following equation [(z+1)(-z^3-5z^2+11z-15)] =0

now how can i factor (-z^3-5z^2+11z-15) ?

i tried it, but i can not solve it... :/
 
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  • #2
Proceed with Ruffini. You'll find another root (because the problem is easy) and the remaining factor is quadratic, whose solutions you get with the formula.
 
  • #3
the solutions are

-1; 3; 1+/-2i

i am going to try with ruffini again.
 
  • #4
i cant. Even knowing the solutions, i can not proceed with ruffini's rule.
Maybe something is escaping me.
 
  • #5
Redo the quotient (z^4-4z^3+6z^2-4z-15)/(z+1), since the leading term must be z^3, not -z^3.
 
  • #6
ok it now makes sense.

now i factor it


(z^3-5z^2+11z-15)/(z-3)



...

but without the solution i would never be able to discover that i should divide (z^3-5z^2+11z-15) by (z-3)
 
  • #7
Do Ruffini again: try with the divisors of -15 of both signs.
 

FAQ: Factoring Polynomial z^4-4z^3+6z^2-4z-15 =0

What is factoring?

Factoring is the process of breaking down a polynomial or algebraic expression into simpler parts, or factors, that when multiplied together, give the original expression.

What is a polynomial?

A polynomial is a mathematical expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. It can have one or more terms and the highest power of the variable is called the degree of the polynomial.

How do I factor a polynomial?

To factor a polynomial, you can use various methods such as grouping, finding common factors, or using the quadratic formula. In this case, we can use the grouping method to factor the given polynomial.

What is the degree of the given polynomial z^4-4z^3+6z^2-4z-15 =0?

The degree of this polynomial is 4, since it is the highest power of the variable z.

How do I solve the equation z^4-4z^3+6z^2-4z-15 =0 by factoring?

First, we can group the first two terms and the last two terms together: (z^4-4z^3) + (6z^2-4z) = 0. Then, we can factor out a common factor of z^3 from the first group and 2z from the second group: z^3(z-4) + 2z(3z-2) = 0. Finally, we can factor out a common factor of (z-4) from both terms: (z-4)(z^3+2z) = 0. This gives us two solutions: z=4 and z=0.

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