Why are there only two roots of this cubic polynomial?

In summary, the conversation discusses finding the roots of a cubic polynomial and explains why there are only two roots instead of three. It is explained that a double root appears visually as two single roots close together and that it behaves differently than a single root. The conversation also mentions the graphical and algebraic ways of proving this.
  • #1
PainterGuy
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Homework Statement
I'm not able to understand why I'm only getting two roots of cubic polynomial.
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Please check my posting.
Hi,

I was trying to find roots of the following cubic polynomial and there are only two roots. I believe there should be three roots. Could you please guide me why there are only two roots?

If you say that the "1" repeats itself as a root then I'd say the same could be said of "0.9". Thank you!
1618828164834.png


Source:
https://www.wolframalpha.com/input/?i=find+roots+x³-2.9x²%2B2.8x-0.9
 
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  • #2
Why do you think the same could be said of the 0.9 root?
 
  • #3
Try factorizing your cubic: [itex]x^3 - 2.9x^2 + 2.8x - 0.9 = (x - 1)(x - 0.9)(x - a)[/itex]. What is [itex]a[/itex]?
 
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  • #4
Thank you!

a=1
 
  • #5
If you look at the graph, at x=1 the graph bounces off the x-axis. This is how you can find double roots visually, if the graph bounces off the axis and doesn't pass through it it's a double root (or a root of multiplicity 4,6,8 etc.The two ways you can prove this to yourself:
One is the sign of the function doesn't change as you pass through a double root but it does as you pass through a single root (you should try to prove this to yourself)

The other more graphical way is that a double root should look a lot like two single roots right next to each other. If you graphed (x-0.9)(x-1.00001)(x-0.99999) you would not visually be able to distinguish those two roots, the graph would go under the x-axis and pop back up over it in a span too small for the naked eye. A double root should look visually the same as that.
 
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  • #6
Office_Shredder said:
If you look at the graph, at x=1 the graph bounces off the x-axis. This is how you can find double roots visually, if the graph bounces off the axis and doesn't pass through it it's a double root (or a root of multiplicity 4,6,8 etc.
I second what Office Shredder has said. Another way to think about this is that near x = 1, the cubic polynomial is acting like a quadratic polynomial; i.e. one whose graph is a parabola.

In factored form, the equation is ##y = (x - .9)(x - 1)^2##. If x is very close to 1, and hence relatively far from .9, the graph of the equation is ##y \approx .1(x - 1)^2##. This graph is a parabola, with vertex at (1, 0) and opening upward. For x values farther away from 1, the x - .9 factor exerts more of an influence.
 
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  • #7
If you examined the graph under a microscope, the double root at x=1 looks and behaves like ##c_1(x-1)^2##, so it touches the y=0 axis and goes back up. Under the same microscope, at the root x=0.9, the graph looks like ##c_2(x-0.9)##, so it goes through the y=0 axis like a straight line. The two situations are not the same at all.
 
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Related to Why are there only two roots of this cubic polynomial?

1. Why can't a cubic polynomial have more than two roots?

A cubic polynomial can have a maximum of three roots, but it is possible for some of these roots to be repeated. This is because a cubic polynomial is a third-degree polynomial, which means it can have a maximum of three solutions when graphed. These solutions are known as roots.

2. How do you know if a cubic polynomial has two roots?

A cubic polynomial will have two roots if it can be factored into two linear factors. This means that the polynomial can be written in the form (ax + b)(cx + d), where a, b, c, and d are constants. If this is the case, the roots will be the values of x that make each factor equal to zero.

3. Why do some cubic polynomials have complex roots?

Complex roots occur when the polynomial cannot be factored into two linear factors with real coefficients. This means that the roots will involve imaginary numbers, such as √-1. Complex roots occur when the graph of the polynomial does not intersect the x-axis at all, or when it intersects at a point that is not a real number.

4. Can a cubic polynomial have only one root?

Yes, a cubic polynomial can have only one root. This occurs when the polynomial can be factored into three linear factors, but two of them are equal. This means that the graph of the polynomial will touch the x-axis at only one point, making that point the only root.

5. How do you find the roots of a cubic polynomial?

The easiest way to find the roots of a cubic polynomial is by factoring it into linear factors. If this is not possible, you can use the cubic formula or the method of completing the square to find the roots. It is also possible to use a graphing calculator or computer software to find the roots numerically.

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