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 4 of 4: Read off the constant term
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f=∏i=18(−ri)=(−12)8=1256f=\prod_{i=1}^8(-r_i)=\left(-\frac12\right)^8=\frac1{256}
Detailed analysis

The constant term is the product of the eight roots with sign (−1)8=1(-1)^8=1. Thus f=∏i=18(−ri)=(−1/2)8=1/256f=\prod_{i=1}^8(-r_i)=(-1/2)^8=1/256, and the polynomial with all roots 1/21/2 shows that this value is attained.