Mastering Factoring 3rd Degree Polynomials: x^3+6x^2-12x+8=0

In summary, factoring is a useful tool for finding the roots or solutions of a 3rd degree polynomial. The general process for factoring includes looking for common factors, using the grouping or trial-and-error method, and using the zero product property. Factoring can also be used for polynomials with coefficients in front of x^3, but the coefficient must be factored out first. If there are no common factors, factoring can still be used, but it may be more difficult to find the roots. Special cases to consider when factoring a 3rd degree polynomial include perfect cubes, imaginary roots, and repeated roots, which require different methods for factoring and finding roots.
  • #1
snoggerT
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-x^3+6x^2-12x+8=0




The Attempt at a Solution



- this is actually part of an eigenvalue problem, but I can't seem to remember how to factor 3rd degree polynomials. Any thing to help me remember would be great.
 
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  • #2
well by investigation we can see that 2 is a root of that equation, now can u perform synthetic division, or divide that polynomial by (x-2).
 
  • #3


As a scientist, it is important to have a strong understanding of mathematical concepts, including factoring polynomials. Factoring is a useful tool in solving equations and understanding the behavior of functions. To factor a 3rd degree polynomial, we must first look for common factors among the terms. In this case, we can factor out a -1 from all terms, giving us -1(x^3-6x^2+12x-8)=0. We can then use the grouping method to factor the remaining polynomial into two binomials, giving us -1[(x^3-6x^2)+(12x-8)]=0. From here, we can factor each binomial separately, giving us -1[x^2(x-6)+4(3x-2)]=0. Finally, we can factor out the common terms, giving us -1(x-2)(x^2+4x-4)=0. This is now in factored form and can be solved by setting each factor equal to zero. I hope this helps you remember the process for factoring 3rd degree polynomials.
 

1. How do I know when to use factoring for a 3rd degree polynomial?

Factoring can be used for any polynomial, but it is especially useful for 3rd degree polynomials because it allows you to easily find the roots or solutions of the equation. If the polynomial has a degree of 3, then it will have 3 roots. Factoring can help you find these roots quickly and efficiently.

2. What is the general process for factoring a 3rd degree polynomial?

The general process for factoring a 3rd degree polynomial is to first look for a common factor among all of the terms. Then, use the grouping method or the trial-and-error method to factor the remaining terms. Finally, use the zero product property to solve for the roots of the equation.

3. Can I use factoring for a 3rd degree polynomial with a coefficient in front of x^3?

Yes, you can still use factoring for a 3rd degree polynomial with a coefficient in front of x^3. The process is the same as factoring a polynomial without a coefficient, except you will need to factor out the coefficient first before using the grouping or trial-and-error method.

4. What if I cannot find any common factors in the 3rd degree polynomial?

If there are no common factors among the terms in the polynomial, then you can still use the grouping or trial-and-error method to factor it. However, it may be more difficult and time-consuming to find the roots of the equation.

5. Are there any special cases to consider when factoring a 3rd degree polynomial?

Yes, there are a few special cases to consider when factoring a 3rd degree polynomial. These include when the polynomial is a perfect cube, when it has imaginary roots, or when it has a repeated root. These cases require different methods for factoring and finding the roots of the equation.

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