Find b so that x - 1 is a factor of x5 + x4 + x3 + x2 + x + b?

Use synthetic division:

1|...1...1...1...1...1...b

..........1....2...3...4...5

---------------------------------

.....1....2...3....4...5| 0

b has to be -5 for the remainder to be 0.

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b is -5. Thank goodness for long division. You can check is by multiplying (x-1) by (x4 + 2x3 + 3x2 + 4x + 5) and you'll get (x5 + x4 + x3 + x2 + x - 5) leaving b as -5.

P(x)=x^5+x^4+x^3+x^2+x+b

if x-1 is a factor P(1) must be zero

P(1)=1+1+1+1+1+b=b+5

equating this to zero we get b=-5

You need to do long division and then change b to the remainder so that the remainder turns out to be zero, thus x-1 becomes a factor of that polynomial.

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