MathLabs

Problem 1

Let p(x)=x8−4x7+7x6+ax5+bx4+cx3+dx2+ex+fp(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f factorise into eight linear factors x−rix-r_i, with ri>0r_i>0 for i=1,2,…,8i=1,2,\ldots,8. Determine all possible values of ff.
Step 3 of 4: A sum of nonnegative squares vanishes
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∑i=18(2ri−1)2=4∑ri2−4∑ri+8=8−16+8=0\sum_{i=1}^8(2r_i-1)^2=4\sum r_i^2-4\sum r_i+8=8-16+8=0
Detailed analysis

Using the two sums, ∑i=18(2ri−1)2=4⋅2−4⋅4+8=0\sum_{i=1}^8(2r_i-1)^2=4\cdot2-4\cdot4+8=0. Every summand is nonnegative because the roots are real, so 2ri−1=02r_i-1=0 for every ii and hence ri=1/2r_i=1/2.