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 2 of 4: Compute the sum of squared roots
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∑i=18ri2=(∑i=18ri)2−2∑i<jrirj=16−14=2\sum_{i=1}^8r_i^2=\left(\sum_{i=1}^8r_i\right)^2-2\sum_{i<j}r_ir_j=16-14=2
Detailed analysis

The identity (∑ri)2=∑ri2+2∑i<jrirj(\sum r_i)^2=\sum r_i^2+2\sum_{i<j}r_ir_j and the preceding values yield ∑i=18ri2=42−2⋅7=2\sum_{i=1}^8r_i^2=4^2-2\cdot7=2.