Factoring a polynomial where factor theorem doesn't work

In summary, the problem is to solve the equation x^3 - 9x^2 + 15x + 30=0 using algebraic methods. The factors of 30 are listed and the use of a graphing calculator is mentioned. A zero close to -1 is found and plugged into the equation, but does not result in a solution. The use of Newton's method and the cubic formula are suggested as potential solutions.
  • #1
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Homework Statement



Solve: x^3 - 9x^2 + 15x + 30

Homework Equations





The Attempt at a Solution



The factors of 30 are +-1, +-2, +-3, +-5, +-6, +-10, +-15, and +-30.

I used my graphing calculator and got a zero close to -1. I plugged it into the original equation and got 5, not 0. I used the zero function on my calculator and found that the zero occurred around -1.13. that's a fraction.

How would I solve this algebraically?
 
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  • #2
Try Newtons method:
Define
[tex] T(x)=x-\frac{f(x)}{f'x}=x-\frac{x^3-9x^2+15x+30}{3x^2+18x+15} [/tex]
 
  • #3
If you are only looking for a numerical approximation, then Newton's method should work fine for you. However, if you are looking for an exact analytical solution, try the cubic formula
 

FAQ: Factoring a polynomial where factor theorem doesn't work

What is the factor theorem and how does it relate to factoring polynomials?

The factor theorem states that if a polynomial has a root or solution, then the corresponding linear factor (x - r) will divide evenly into the polynomial. This makes it easier to factor polynomials, as we can use the roots to find the linear factors and then use those factors to find the remaining factors.

In what cases does the factor theorem not work when factoring polynomials?

The factor theorem will not work if the polynomial does not have any roots or if the roots are irrational or complex numbers. In these cases, we cannot use the roots to find the linear factors and will need to use other methods to factor the polynomial.

What are some other methods for factoring polynomials when the factor theorem doesn't work?

Some other methods for factoring polynomials include grouping, using the difference of squares formula, and using the quadratic formula for quadratics. These methods can be used when the roots are not easily identifiable or when the polynomial does not have any real roots.

Can a polynomial have no real roots and still be factored?

Yes, a polynomial can have no real roots and still be factored. This is possible because of the fundamental theorem of algebra, which states that every polynomial of degree n has n complex roots. So even if a polynomial does not have any real roots, it can still be factored into linear and/or quadratic factors using complex numbers.

Is there a way to check if a polynomial can be factored using the factor theorem?

Yes, there is a way to check if a polynomial can be factored using the factor theorem. If a polynomial is in the form of ax^n + bx^(n-1) + ... + c, where a is the leading coefficient and c is the constant term, then we can check if the polynomial has any roots by finding the possible rational roots using the rational root theorem. If the polynomial has any rational roots, then we can use the factor theorem to factor it. If it does not have any rational roots, then we will need to use other methods to factor the polynomial.

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