Meaning of multi zero solutions

[Atomised]

Homework Statement

I have solved the equation $$x^3=7x^2$$ giving roots at $$x=7, x=0$$ The solutions in the book also give a specific third solution of $$x=0$$ again.

I can't see the point of this unless it is to reassure the reader that there are no further solutions to look for given that it is widely known that cubics have three solutions. Do they mean +0 and -0? Whatever that means.

Examining the graph of the above function I see intersections with the x-axis ONCE at x=0 and once at x=7, I have seen cubics which intersect the x-axis three times but this is not one of them. What does the second zero signify?

Many thanks

 

[Mark44]

Homework Statement

I have solved the equation $$x^3=7x^2$$ giving roots at $$x=7, x=0$$ The solutions in the book also give a specific third solution of $$x=0$$ again.

I can't see the point of this unless it is to reassure the reader that there are no further solutions to look for given that it is widely known that cubics have three solutions. Do they mean +0 and -0? Whatever that means.

I guess all they're doing is emphasizing the fact that the root at 0 has multiplicity 2. No, they don't mean +0 and -0, which mathematically are the same thing. (In computers, floating point units can store -0 and +0 differently, but there's no difference computationally.)

Examining the graph of the above function I see intersections with the x-axis ONCE at x=0 and once at x=7, I have seen cubics which intersect the x-axis three times but this is not one of them. What does the second zero signify?

Many thanks

I assume you are graphing y = x³ - 7x²The curve looks different at the root with mult. 2, as compared to how it looks at the other root (7). Near x = 0, the graph has a sort of parabolic shape (that opens downward). Near x = 7, the graph is nearly a straight line, with a positive slope. Knowing the multiplicity of the roots can give you a better idea of the shape of the graph.

 

[Atomised]

I guess all they're doing is emphasizing the fact that the root at 0 has multiplicity 2. No, they don't mean +0 and -0, which mathematically are the same thing. (In computers, floating point units can store -0 and +0 differently, but there's no difference computationally.)

I assume you are graphing y = x³ - 7x²The curve looks different at the root with mult. 2, as compared to how it looks at the other root (7). Near x = 0, the graph has a sort of parabolic shape (that opens downward). Near x = 7, the graph is nearly a straight line, with a positive slope. Knowing the multiplicity of the roots can give you a better idea of the shape of the graph.

How intriguing - what does a root having a multiplicity of two actually mean? Can a root have a multiplicity >2?

Yes I see what you mean, the x-axis is kissed by the parabola but does not cross it in that region.
So that happens generally when you get coincident roots?





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[Mark44]

The function y = f(x) = (x - 1)²(x - 7) is a cubic polynomial in factored form. This form makes it easy to find the x-intercepts, which in this case are at x = 1 (multiplicity 2) and x = 1 (multiplicity 1). Multiplicity of a root x = a tells you how many times a factor (x - a) appears in the factored form of the polynomial. The multiplicity of a root plays a big role in the shape of the graph of the function at points near that root.

The graph of this function is similar to the graph of the function in post #1 in this thread.

Another function, y = g(x) = (x - 1)³(x - 7) is a quartic (fourth degree) polynomial, also in factored form. The intercepts are again at x = 1 and x = 7, but this time the multiplicity of the root at x = 1 is 3. When x is near 1, the graph of g has a shape similar to the graph of y = -x³, sort of an S shape. To the left of x = 1, the y values of the graph of g are positive; to the right of x = 1, the y values are negative. At some point between 1 and 7, the graph turns back up and crosses the x-axis at x = 7.

 

[HallsofIvy]

Another way of looking at it is that solving $x^3= 7x^2$ is the same as solving the polynomial equation $x^3- 7x^2= x^2(x- 7)= (x- 0)(x- 0)(x- 7)= 0$.

Yes, of course an equation can have roots of multiplicity greater than 2: $x^4= 7x^3$ is equivalent to x^3(x- 7)= (x- 0)(x- 0)(x- 0)(x- 7)= 0.

 

[Atomised]

Love the $$(x-0)$$ stuff thanks.







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