SOLUTION: If f is a polynomial of degree 4 such that f(0) = f(1) = f(2) = f(3) = 1 and f(4) = 0, then determine f(5).

Algebra ->  Functions -> SOLUTION: If f is a polynomial of degree 4 such that f(0) = f(1) = f(2) = f(3) = 1 and f(4) = 0, then determine f(5).      Log On


   

Question 1077691: If f is a polynomial of degree 4 such that
f(0) = f(1) = f(2) = f(3) = 1
and
f(4) = 0,
then determine f(5).
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29+=+ax%5E4+%2B+bx%5E3+%2B+cx%5E2+%2B+dx+%2B+e+
+f%280%29+=+1+=+0%2B0%2B0%2B0%2Be+ —> +e+=+1+ (use e=1 in evaluating f(1)…f(4) below)
+f%281%29+=+1+ —> +a+%2B+b+%2B+c+%2B+d+%2B+e+=+1+ —> a+%2B+b+%2B+c+%2B+d+=+0+
+f%282%29+=+1+ —> +a%2816%29+%2B+b%288%29+%2B+c%284%29+%2B+d%282%29+%2B+e+=+1+ —> +16a+%2B+8b+%2B+4c+%2B+2d+=+0+
+f%283%29+=+1+ —> ... —> +81a+%2B+27b+%2B+9c+%2B+d+=+0+
+f%284%29+=+0+ —> … —> +256a+%2B+64b+%2B+16c+%2B+4d+%2B+1+=+0+
—
This generates the following matrix (moving constants to right hand side):
—

The inverse of the 4x4 matrix is (from https://www.symbolab.com/solver/matrix-inverse-calculator )

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And the solution we seek is:




+a+=+-1%2F6+%2B+1%2F4+-+1%2F6+-+1%2F24+=+-3%2F24+
+b+=+3%2F2+-+2+%2B+7%2F6+%2B+1%2F4+=+22%2F24+
+c+=+-57%2F24+
+d+=+62%2F24+

pull out 1/24:
f%28x%29+=+%281%2F24%29%28-3x%5E4+%2B+22x%5E3+-57x%5E2+%2B+62x+%2B+24%29+
—
Now we are ready to find f(5):
+f%285%29+=+%281%2F24%29%28-3%28625%29+%2B+22%28125%29+-57%2825%29+%2B+62%285%29+%2B+24%29+
+f%285%29+=+%281%2F24%29%28-216%29+=+highlight%28-9%29+