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Fast Fourier Transform (FFT) (20 points) In this question, we'll see a basic example of using FFT to multiply two polynomials. Our polynomials here will be $f(x) = 2+x$ and $g(x)=1-x+x^2$. Recall that $i = \sqrt{-1}$, which means that the 4th roots of unity are 1, i, -1, and -i (each of these complex numbers raised to the 4th power is 1). Show your work for all parts. Part A (10 points) Compute the FFT (i.e. the \"value representation\" ) of $f(x)$ and $g(x)$ by evaluating them at the 4th roots of unity. You can just leave your answer as a tuple of complex numbers, for example, the FFT of $1 + x$ would be $(2, 1+i, 0, 1-i)$. (No need to do the actual recursive method, simply evaluating by hand is fine) Part B (10 points) Now, take the element-wise product of the two FFTs you computed above (that is, multiply element 1 of FFT($f(x)$) with element 1 of FFT($g(x)$), element 2 with element 2, etc.). Then, separately, compute the product of the two polynomials, $h(x) = f(x)g(x)$ (just with normal polynomial multiplication). Compute the FFT of $h(x)$. It should match the element-wise product from earlier!

          Fast Fourier Transform (FFT) (20 points)
In this question, we'll see a basic example of using FFT to multiply two polynomials. Our polynomials here
will be $f(x) = 2+x$ and $g(x)=1-x+x^2$. Recall that $i = \sqrt{-1}$, which means that the 4th roots of unity are 1,
i, -1, and -i (each of these complex numbers raised to the 4th power is 1). Show your work for all parts.
Part A (10 points)
Compute the FFT (i.e. the \"value representation\" ) of $f(x)$ and $g(x)$ by evaluating them at the 4th roots of
unity. You can just leave your answer as a tuple of complex numbers, for example, the FFT of $1 + x$ would
be $(2, 1+i, 0, 1-i)$. (No need to do the actual recursive method, simply evaluating by hand is fine)
Part B (10 points)
Now, take the element-wise product of the two FFTs you computed above (that is, multiply element 1 of
FFT($f(x)$) with element 1 of FFT($g(x)$), element 2 with element 2, etc.). Then, separately, compute the
product of the two polynomials, $h(x) = f(x)g(x)$ (just with normal polynomial multiplication). Compute
the FFT of $h(x)$. It should match the element-wise product from earlier!

Fast Fourier Transform (FFT) (20 points)
In this question, we'll see a basic example of using FFT to multiply two polynomials. Our polynomials here
will be f(x) = 2+x and g(x)=1-x+x^2. Recall that i = √(-1), which means that the 4th roots of unity are 1,
i, -1, and -i (each of these complex numbers raised to the 4th power is 1). Show your work for all parts.
Part A (10 points)
Compute the FFT (i.e. the v̈alue representation)̈ of f(x) and g(x) by evaluating them at the 4th roots of
unity. You can just leave your answer as a tuple of complex numbers, for example, the FFT of 1 + x would
be (2, 1+i, 0, 1-i). (No need to do the actual recursive method, simply evaluating by hand is fine)
Part B (10 points)
Now, take the element-wise product of the two FFTs you computed above (that is, multiply element 1 of
FFT(f(x)) with element 1 of FFT(g(x)), element 2 with element 2, etc.). Then, separately, compute the
product of the two polynomials, h(x) = f(x)g(x) (just with normal polynomial multiplication). Compute
the FFT of h(x). It should match the element-wise product from earlier!

Added by Belen F.

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