Solving for X Intercepts: How to Find the Real Answers for x^3-3x^2+3

  • Thread starter Whalstib
  • Start date
In summary: The problem with using the quadratic formula is that it's not in the form of ax^2+bx+c. You could try the Rational Roots Theorem, but that's not going to work in this case because all three roots are irrational. There's the Cubic Formula, but it's very complicated to use. Outside of those two methods, you're not going to be able to solve this algebraically.The Durand Kerner method might work, but it's not guaranteed.
  • #1
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Homework Statement


Find x intercept of:

x^3-3 x^2+3


Homework Equations



I can clearly see three x intercepts on the graph but can not determine them mathematically. Quadratic formula and calculator give non real answer. Book says: ( -.879, 0) (1.347,0) (2.5232, 0)


The Attempt at a Solution



I am totally stuck as the quadratic gives a negative discriminant and a non real answer... yet there are indeed three intercepts.

What am I doing wrong?

Thanks,

Warren
 
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  • #2
I don't think you could use the quad. formula here because it's not in the form of ax^2+bx+c. Did you go over division? I believe you have to use long division to find the zeroes.
 
  • #3
stratusfactio said:
I don't think you could use the quad. formula here because it's not in the form of ax^2+bx+c.
I agree.
Well, I don't know about finding exact roots, but you can get a general idea of where the zeroes are by plugging in a few numbers. For example, with x=1 you get a positive answer, but with x=-1 you get a negative answer. So in between x=1 and x=-1 there is a root.
 
  • #4
stratusfactio said:
I don't think you could use the quad. formula here because it's not in the form of ax^2+bx+c. Did you go over division? I believe you have to use long division to find the zeroes.

D'oh! Of course! Man, it is possible to study too long...! I'm still at a loss...

I'm not sure how to work the division...any hints?...

Thanks,

Warren
 
  • #5
There's the Rational Roots Theorem, but it's not going to work in this case, because you can see that all three roots are irrational. There's the Cubic Formula, but it's very complicated to use. Outside of those two methods, you're not going to be able to solve this algebraically.
 
  • #6
Durand Kerner method?
 
  • #7
So...

Use the calculator and zero function to estimate is all that could possibly be expected from a calc 1 student right?
I got to say this is so typical of the Larson Calculus of a Single Variable book! Each chapter has problems that are near impossible to solve with the instruction given to that point.

When learning the product rule several problems were given that demanded quotient rule and the answer book gave the the answer as quotient rule form. Unfortunatly quotient rule isn't presented till the next section!

Thanks,

Warren
 
  • #8
I have a friend who took college algebra for the extra A (he tested out of it) and he swears they had to solve cubic equations.

But... here you go.

cubic.gif
 

What is the formula for finding the x-intercepts of a polynomial equation?

The formula for finding the x-intercepts of a polynomial equation is x = -b/a, where a is the coefficient of the x^2 term and b is the coefficient of the x term.

How do you solve for the x-intercepts of x^3-3x^2+3?

To solve for the x-intercepts of x^3-3x^2+3, you can first factor out an x^2 term to get x^2(x-3)+3. Then, set each factor equal to zero and solve for x. This will give you the two real solutions for x, which are -1 and 3.

Can there be more than two x-intercepts for a polynomial equation?

Yes, there can be more than two x-intercepts for a polynomial equation. This can happen when the polynomial has a degree higher than 3, such as x^4-3x^3+2x^2+5x-6. In this case, there can be up to four x-intercepts.

Can the x-intercepts of a polynomial equation be imaginary numbers?

Yes, the x-intercepts of a polynomial equation can be imaginary numbers. This can happen when the polynomial has complex roots, which occur when the discriminant of the quadratic formula is negative. For example, the polynomial x^2+4x+5 has two imaginary x-intercepts of -2i and 2i.

What is the significance of finding the x-intercepts of a polynomial equation?

Finding the x-intercepts of a polynomial equation can help us understand the behavior of the graph of the polynomial. The x-intercepts represent the points where the graph crosses the x-axis, and can tell us the roots or solutions of the equation. They can also help us determine the maximum or minimum values of the polynomial, and whether the graph is increasing or decreasing in certain intervals.

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