Factoring a difficult polynomial so that I can extend equation

In summary, the extended function that is continuous at x=2 can be obtained by factoring the numerator and denominator, where the factors of the numerator are (x-5)(x-2)(x+3). Using the rational roots theorem, it can be determined that (x-2) is a factor, and dividing the numerator by (x-2) results in a quadratic that can then be factored to obtain the other two factors.
  • #1
pugola12
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Homework Statement


Give a formula for the extended function that is continuous at the indicated point.

(x3-4x2-11x+30)/(x2-4), x=2


Homework Equations





The Attempt at a Solution


I know that I have to factor the top and bottom so that I can cancel terms that cause the function to be undefined at x=2, and I know how to do everything after that. However, I'm stuck on how to factor x3-4x2-11x+30. I know that the factors are (x-5)(x-2)(x+3), but I don't know how to go about getting there. I tried grouping, but it didn't work since there is no way that I see to factor -11x+30.
x3-4x2-11x+30

x2(x-4)-11x+30

I tried grouping other ways, but they did not work either.
 
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  • #2
In general, what you want is the rational roots theorem. If (x-r) is a factor of the numerator then r must divide 30. That gives a limited number of guesses for possible factors. But in this particular case, it's a pretty good guess that (x-2) is factor. Once you know that, divide the numerator by (x-2) to get a quadratic. Then factor the quadratic to get the other two factors.
 
  • #3
Oh gee, I completely forgot about that. (It's been a while since Algebra II) That makes total sense. Thank you so much for your help!
 

Related to Factoring a difficult polynomial so that I can extend equation

1. What is factoring a polynomial?

Factoring a polynomial is the process of breaking down a polynomial into its simpler components or factors. This is done by finding the common factors and simplifying the polynomial into a product of those factors.

2. Why is factoring a polynomial important?

Factoring a polynomial is important because it can help us solve equations and understand the behavior of a polynomial function. It also allows us to simplify complex expressions and identify patterns in data.

3. How do I factor a difficult polynomial?

Factoring a difficult polynomial can be done by using various methods such as grouping, factoring by grouping, factoring by grouping and grouping different terms, and the quadratic formula. It is important to identify the common factors and use the appropriate method to factor the polynomial.

4. Can factoring a polynomial help me extend an equation?

Yes, factoring a polynomial can help extend an equation by simplifying it into a product of factors. This can help us identify patterns and relationships between different variables in the equation, and potentially lead to new insights and discoveries.

5. What are some tips for factoring a difficult polynomial?

Some tips for factoring a difficult polynomial include:

  • Identify common factors and use them to simplify the polynomial
  • Use the appropriate method based on the degree and structure of the polynomial
  • Check your answer by expanding the factored form to ensure it is equivalent to the original polynomial
  • Practice and familiarize yourself with different factoring techniques

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