Behavior Near Zero For f(x)

[mathdad]

Say f(x) = x^2(x - 2). What is the behavior of this polynomial function near zero? I have an idea about how to do this but seeking solution steps here. Thanks.

 

[pka]

Say f(x) = x^2(x - 2). What is the behavior of this polynomial function near zero? I have an idea about how to do this but seeking solution steps here.

When you say "polynomial function near zero" do you mean \(\displaystyle f(x)\approx 0\) OR do you mean \(\displaystyle x\approx 0~?\)
Now you may object that those to are not different. But consider this function \(\displaystyle g(x)=(x^2+1)(x-1)(x+2)\).
If \(\displaystyle x\approx 0\) then \(\displaystyle g(x)\approx -2\)
If \(\displaystyle x\approx 1\) then \(\displaystyle g(x)\approx 0\).
If \(\displaystyle g(x)\approx 0\) then \(\displaystyle x\approx -2\text{ OR }x\approx 1\)
I truly do hope that you see the ambiguity in the wording of your post.

 

[MarkFL]

If we're talking about the behavior of the function near \(x=0\). then we see that this is a zero of the function, a polynomial function, and that the root is of even multiplicity, which means the function will be tangent to the \(x\)-axis there, and we can see the function is negative on either side of this zero.

 

[Steven G]

Say f(x) = x^2(x - 2). What is the behavior of this polynomial function near zero? I have an idea about how to do this but seeking solution steps here. Thanks.

Assuming you mean x~0 and you are not very good at these type problems then you need to get your hands a little dirty!
Compute f(-.5)
Compute f(-.1)
Compute f(-.01)
Compute f(-.0001)
Now Compute f(.5)
Compute f(.1)
Compute f(.01)
Compute f(.0001)

Can you answer the question now? Isn't this what your instructor and/or book mentioned?

 

[Dr.Peterson]

Say f(x) = x^2(x - 2). What is the behavior of this polynomial function near zero?

I will assume you meant to say, "What is the behavior of this polynomial function near each zero?" If not, that's the more interesting question anyway.

Here's a nice way to see the behavior of a polynomial (in factored form) near a zero (that is, any place where f(x) = 0):



The purple curve is your function.

One zero is at [MATH]x=0[/MATH], where the first factor is zero. I've replaced x in the other factor with 0, to hold that factor's value fixed while we let x vary in the first factor. That produces the red curve, which is a parabola, [MATH]y = -2x^2[/MATH]. That curve shows how the function behaves near that zero: Like any parabola, it touches the axis and turns around, having the same sign on both sides. You can tell that just by seeing that the factor is [MATH]x^2[/MATH] (multiplicity 2); any even power never changes sign, so such a factor doesn't change the sign of the function neat its zero.

The other zero is at x=2, where the second factor is zero. The blue curve shows what happens when we replace x with 2 in the first factor, so only the second factor affects the curve. This function, [MATH]y = 4(x-2)[/MATH], is a linear function, and its graph is a straight line tangent to the given curve. So at that point, the curve behaves like any linear function, cutting through the axis and changing sign. That is true of any zero with multiplicity 1 (that is, corresponding to a factor with exponent 1), because that factor does change sign.

Of course, you don't need to even think about the graphs I've drawn; they just help show the conclusions. All you need to do is to look at the multiplicity of each zero, which is the degree of its factor.

 

[mathdad]

When you say "polynomial function near zero" do you mean \(\displaystyle f(x)\approx 0\) OR do you mean \(\displaystyle x\approx 0~?\)
Now you may object that those to are not different. But consider this function \(\displaystyle g(x)=(x^2+1)(x-1)(x+2)\).
If \(\displaystyle x\approx 0\) then \(\displaystyle g(x)\approx -2\)
If \(\displaystyle x\approx 1\) then \(\displaystyle g(x)\approx 0\).
If \(\displaystyle g(x)\approx 0\) then \(\displaystyle x\approx -2\text{ OR }x\approx 1\)
I truly do hope that you see the ambiguity in the wording of your post.

The wording is not my own.

 

[mathdad]

I will assume you meant to say, "What is the behavior of this polynomial function near each zero?" If not, that's the more interesting question anyway.

Here's a nice way to see the behavior of a polynomial (in factored form) near a zero (that is, any place where f(x) = 0):

View attachment 12070

The purple curve is your function.

One zero is at [MATH]x=0[/MATH], where the first factor is zero. I've replaced x in the other factor with 0, to hold that factor's value fixed while we let x vary in the first factor. That produces the red curve, which is a parabola, [MATH]y = -2x^2[/MATH]. That curve shows how the function behaves near that zero: Like any parabola, it touches the axis and turns around, having the same sign on both sides. You can tell that just by seeing that the factor is [MATH]x^2[/MATH] (multiplicity 2); any even power never changes sign, so such a factor doesn't change the sign of the function neat its zero.

The other zero is at x=2, where the second factor is zero. The blue curve shows what happens when we replace x with 2 in the first factor, so only the second factor affects the curve. This function, [MATH]y = 4(x-2)[/MATH], is a linear function, and its graph is a straight line tangent to the given curve. So at that point, the curve behaves like any linear function, cutting through the axis and changing sign. That is true of any zero with multiplicity 1 (that is, corresponding to a factor with exponent 1), because that factor does change sign.

Of course, you don't need to even think about the graphs I've drawn; they just help show the conclusions. All you need to do is to look at the multiplicity of each zero, which is the degree of its factor.

You hit the nail on the head! Perfect.

 

[Dr.Peterson]

What is the behavior of this polynomial function near zero?

The wording is not my own.

Interesting. If what you asked is exactly what the book said, then it is asking only about behavior near x=0, not at zeroes of the function. That's not a question I've seen in textbooks as far as I can recall; it would probably be asked only within a larger context asking about each zero in turn.

I have a Sullivan Precalculus book, but I don't find such wording there, so I can't check what they mean.

 

[mathdad]

Interesting. If what you asked is exactly what the book said, then it is asking only about behavior near x=0, not at zeroes of the function. That's not a question I've seen in textbooks as far as I can recall; it would probably be asked only within a larger context asking about each zero in turn.

I have a Sullivan Precalculus book, but I don't find such wording there, so I can't check what they mean.

I have a Sullivan college algebra textbook 9th edition.