Determine End Behavior of polynomial
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Re: Polynomials of least degree with given zeros
We'd have a MUCH better idea of how to help you if you would show us how you started, even if you think it is "completly wrong".
Here's a hint....
if x = a is a zero of a given polynomial, then x - a is a FACTOR of that polynomial.
If you are given three zeros, like a, b, and c, then you're also given three FACTORS of that polynomial. If a is a zero, then x - a is a factor. If b is a zero, then x - b is a factor. If c is a zero, then x - c is a factor.
So, if you have three zeros (a, b, and c), then the simplest polynomial having these zeros would be one with factors of (x - a)(x - b)(x - c).
See if you can apply this to your problem. If you're still having trouble, please repost and show us ALL of your work.
enya12 said:
We'd have a MUCH better idea of how to help you if you would show us how you started, even if you think it is "completly wrong".
Here's a hint....
if x = a is a zero of a given polynomial, then x - a is a FACTOR of that polynomial.
If you are given three zeros, like a, b, and c, then you're also given three FACTORS of that polynomial. If a is a zero, then x - a is a factor. If b is a zero, then x - b is a factor. If c is a zero, then x - c is a factor.
So, if you have three zeros (a, b, and c), then the simplest polynomial having these zeros would be one with factors of (x - a)(x - b)(x - c).
See if you can apply this to your problem. If you're still having trouble, please repost and show us ALL of your work.
Re: Polynomials of least degree with given zeros
Roots are (per your problem) 3, 2i, -2i
So: (x - 3)(x - 2i)(x + 2i)
If you don't know that 2i times -2i = 4, then you're not ready for this; see your teacher.
WHERE do you get (x-2)?enya12 said:
Roots are (per your problem) 3, 2i, -2i
So: (x - 3)(x - 2i)(x + 2i)
If you don't know that 2i times -2i = 4, then you're not ready for this; see your teacher.
Re: Polynomials of least degree with given zeros
i cant get help from her becuase she gave us papers to be able to redo the problems we missed on the test we just took. ok i understand what you put down now we have to foil right? or distribute... etc.
so its : x^3-3x^2+4x-12
thats it.
i cant get help from her becuase she gave us papers to be able to redo the problems we missed on the test we just took. ok i understand what you put down now we have to foil right? or distribute... etc.
so its : x^3-3x^2+4x-12
thats it.
Determine End Behavior of Polynomials
Determine the end behavior of the following polynomial. this was multiple choice and i put B. here it is: but for some reason would it be C ?
F(x)=2x^4-3x^3-5x+6
a. x---->infinite ;f(x)----->infinite
x----> negative infinite; f(x)---->negative infinite
b. x--->infinite; f(x)--->negative infinite
x--->negative infinite; f(x)--->infinite
c. x--->infinite; f(x)---> negative infinite
x--->negative infinite; f(x)--->negative infinite
d. x--->infinite; f(x)--->infinite
x--->negative infinite; f(x)---> infinite
Determine the end behavior of the following polynomial. this was multiple choice and i put B. here it is: but for some reason would it be C ?
F(x)=2x^4-3x^3-5x+6
a. x---->infinite ;f(x)----->infinite
x----> negative infinite; f(x)---->negative infinite
b. x--->infinite; f(x)--->negative infinite
x--->negative infinite; f(x)--->infinite
c. x--->infinite; f(x)---> negative infinite
x--->negative infinite; f(x)--->negative infinite
d. x--->infinite; f(x)--->infinite
x--->negative infinite; f(x)---> infinite
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It's the leading term that determines global behavior.
In other words, we look at the degree of the polynomial and the sign on the leading coefficient.
Function F has degree 4 because the leading term is 2x^4.
x^4 is the same as x^2 * x^2.
You should know that the square of any number (other than zero) is always positive.
Therefore, x^2 * x^2 will be two positive numbers multiplied together, regardless of whether x is huge positively or negatively.
What sign do you get on a product, when two positive numbers are multiplied together? Positive!
And, the leading coefficient 2 is also positive.
So, there is nothing to cause function F to head toward negative infinity.
The general shape of any polynomial with even degree and positive leading coefficient opens upward. In the case of 2nd-degree (quadratic) polynomials, the shape is a parabola, and for even powers larger than 2 the general shape is like a big U, when you zoom out.
It's the leading term that determines global behavior.
In other words, we look at the degree of the polynomial and the sign on the leading coefficient.
Function F has degree 4 because the leading term is 2x^4.
x^4 is the same as x^2 * x^2.
You should know that the square of any number (other than zero) is always positive.
Therefore, x^2 * x^2 will be two positive numbers multiplied together, regardless of whether x is huge positively or negatively.
What sign do you get on a product, when two positive numbers are multiplied together? Positive!
And, the leading coefficient 2 is also positive.
So, there is nothing to cause function F to head toward negative infinity.
The general shape of any polynomial with even degree and positive leading coefficient opens upward. In the case of 2nd-degree (quadratic) polynomials, the shape is a parabola, and for even powers larger than 2 the general shape is like a big U, when you zoom out.