Finding x-intercept of cubic functions? / Factoring cubic functions

In summary, the conversation discusses the difficulty of dealing with cubic functions and finding their roots and x-intercepts. The suggested solution is to use factoring, either by trial and error or by applying the Rational Root Theorem. The conversation also mentions the use of factoring by grouping and recommends practicing with simpler factorization problems.
  • #1
liquidwater
11
0

Homework Statement


My problem is that whenever I have to use cubic functions - whether it's finding the roots (when dy/dx kis a cubic function) or finding the x-intercepts... With quadratic functions I usually use the quadratic formula. I try to figure out how to factor these equations in my head but I can rarely get it right.

Right now my problem is with finding the x-intercepts of
y = 2x3 - 3x2 - 12x

Homework Equations



Finding the local maximum/minimum when dy/dx = 4x3 + 12x2 - 4x - 12

The Attempt at a Solution


I've tried using Newton's method on the relevant equation question, but that failed. x1 equalled -3 and x2 equalled 109, using an initial guess of 1. I won't put working because I think this really isn't what I'm supposed to do.Don't really need answers to these questions, just really need some help on how to deal with cubic functions. Thanks a lot if anyone can help.
 
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  • #2
Hi liquidwater,

To the first problem, you can circumvent dealing with a cubic altogether by factoring.

For the second one, try factoring by grouping.

I hope this helps!
 
  • #3
tjackson3 said:
Hi liquidwater,

To the first problem, you can circumvent dealing with a cubic altogether by factoring.

For the second one, try factoring by grouping.

I hope this helps!

Thanks for replying,

That's what I'm really having trouble with... "With quadratic functions I usually use the quadratic formula. I try to figure out how to factor these equations in my head but I can rarely get it right." I can't think of the right combination of numbers to use.
 
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  • #4
liquidwater said:
Thanks for replying,

That's what I'm really having trouble with... "With quadratic functions I usually use the quadratic formula. I try to figure out how to factor these equations in my head but I can rarely get it right." I can't think of the right combination of numbers to use.

Then you should spend some extra time going over factorization problems. Not being able to do simple factorization problems correctly will be a deterrent to getting the nuts and bolts of more complex problems.
 
  • #5
Mark44 said:
Then you should spend some extra time going over factorization problems. Not being able to do simple factorization problems correctly will be a deterrent to getting the nuts and bolts of more complex problems.

Thanks... I can't factorize cubic functions, I should have explicitly asked for help with that.

Should I be able to this kind of thing in my head, just by looking at it? Or is there some sort of formulae/tricks that I can use?

I can factorize quadratics fine (but I often just turn to quadratic formula).
 
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  • #6
The first one is pretty easy, since each term has a factor of x.
2x3 - 3x2 - 12x = x(2x2 - 3x - 12).
Now it's just a matter of factoring 2x2 - 3x - 12.

Pretty obviously, the factorization is going to look like (2x + ?)(x + ?). Now it's a matter of trial and error of picking factors of -12 that combine to give you a middle term of -3x. In factoring -12, one factor will have to be negative and one positive.

For the other polynomial, your derivative, that's going to be a bit harder, but you can start by noticing that each coefficient is a multiple of 4.
4x3 + 12x2 - 4x - 12 = 4(x3 + 3x2 - x - 3 ).

One approach that works in special cases is to notice x3 + 3x2 and - x - 3 can be factored by a method called factoring by grouping, already mentioned by tjackson3.

Another more general approach is to use the Rational Root Theorem to see if the cubic has any roots that are rational numbers. If p/q is a root of the cubic, then one factor is (x - p/q).

This theorem says that if p/q is a root of anxn + an-1xn-1 + ... + a1x + a0, then p has to divide a0, and q has to divide an.

For this cubic, x3 + 3x2 - x - 3, if p/q is a root, then p has to divide -3 and q has to divide 1. This means that the possible roots are +/-3 or +/-1, or equivalently, that x + 3, x -3, x + 1, or x - 1 are the possible linear factors. Each of these can be tested using long polynomial division or (easier) synthetic division.
 
  • #7
Mark44 said:
The first one is pretty easy, since each term has a factor of x.
2x3 - 3x2 - 12x = x(2x2 - 3x - 12).
Now it's just a matter of factoring 2x2 - 3x - 12.

Pretty obviously, the factorization is going to look like (2x + ?)(x + ?). Now it's a matter of trial and error of picking factors of -12 that combine to give you a middle term of -3x. In factoring -12, one factor will have to be negative and one positive.

For the other polynomial, your derivative, that's going to be a bit harder, but you can start by noticing that each coefficient is a multiple of 4.
4x3 + 12x2 - 4x - 12 = 4(x3 + 3x2 - x - 3 ).

One approach that works in special cases is to notice x3 + 3x2 and - x - 3 can be factored by a method called factoring by grouping, already mentioned by tjackson3.

Another more general approach is to use the Rational Root Theorem to see if the cubic has any roots that are rational numbers. If p/q is a root of the cubic, then one factor is (x - p/q).

This theorem says that if p/q is a root of anxn + an-1xn-1 + ... + a1x + a0, then p has to divide a0, and q has to divide an.

For this cubic, x3 + 3x2 - x - 3, if p/q is a root, then p has to divide -3 and q has to divide 1. This means that the possible roots are +/-3 or +/-1, or equivalently, that x + 3, x -3, x + 1, or x - 1 are the possible linear factors. Each of these can be tested using long polynomial division or (easier) synthetic division.

:approve: Thanks a lot! I'll start working on this stuff.
 

What is the general method for finding the x-intercept of a cubic function?

To find the x-intercept of a cubic function, you need to set the function equal to zero and then factor it. The x-intercept(s) will be the solutions to the factored equation.

What do I do if the cubic function cannot be easily factored?

If the cubic function cannot be easily factored, you can use the Rational Root Theorem and the synthetic division method to find the x-intercepts. This involves testing possible rational roots and using synthetic division to determine which ones are actual roots.

How do I know if the x-intercept(s) are real or complex numbers?

The x-intercept(s) of a cubic function can be real or complex numbers. If the discriminant (b^2-4ac) of the quadratic equation formed by the first two terms of the factored cubic function is positive, the x-intercept(s) will be real numbers. If the discriminant is negative, the x-intercept(s) will be complex numbers.

Can I use the quadratic formula to find the x-intercept(s) of a cubic function?

No, the quadratic formula can only be used to find the solutions for quadratic equations. To find the x-intercept(s) of a cubic function, you need to factor or use the Rational Root Theorem and synthetic division.

Is there a specific order in which I should factor a cubic function?

Yes, there is a specific order in which you should factor a cubic function. First, you should check for any common factors among all three terms. Then, you should look for any difference of squares or perfect cubes. Finally, you can use the grouping method or the sum/difference of cubes formula to factor the remaining terms.

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