Find a third-degree polynomial function with real coefficients and with zeros at -4 and -3 5i
1 answer:
If -3+5i is a solution, then by the conjugate root theorem, -3-5i is also a solution. We find the polynomial by multiplying together the factors. If x = -4, then x + 4 is the factor. If x = -3+5i, then (x-(-3+5i)) is a factor, and so is (x-(-3-5i)). Simplifying those down gives us as the first factor as (x+3-5i) and the second as (x+3+5i). We can FOIL those 2 together to get their product, and then FOIL in x+4. FOILing the 2 complex factors together gives us . If we combine like terms and cross out things that cancel it's much easier than what it looks like there! It simplifies down to
. Since i^2 = -1, it simplifies further to
and finally, to
. Now we will FOIL in x+4.
. Our final simplified third degree polynomial is
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Answer:
Parent function is compressed by a factor of 3/4 and shifted to right by 3 units.
Step-by-step explanation:
We are asked to describe the transformation of function as compared to the graph of
.
We can write our transformed function as:
Now let us compare our transformed function with parent function.
Let us see rules of transformation.
,
,
Scaling of a function:
If a>1 , so function is stretched vertically.
If 0<a<1 , so function is compressed vertically.
As our parent function is multiplied by a scale factor of 3/4 and 3/4 is less than 1, so our parent function is compressed vertically by a factor of 3/4.
As 3 is being subtracted from x, so our parent function is shifted to right by 3 units or a horizontal shift to right by 3 units.
Therefore, our parent graph is compressed by a factor of 3/4 and shifted to right by 3 units to get our new graph.