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 1 of 4: Apply Vieta's formulas
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∑i=18ri=4,∑1≤i<j≤8rirj=7\sum_{i=1}^8 r_i=4,\qquad \sum_{1\le i<j\le8}r_ir_j=7
Detailed analysis

Write the roots as r1,…,r8r_1,\ldots,r_8. Comparing the coefficients of x7x^7 and x6x^6 gives ∑i=18ri=4\sum_{i=1}^8 r_i=4 and ∑1≤i<j≤8rirj=7\sum_{1\le i<j\le8}r_ir_j=7.