Question 737612: Solve. Remember to use the Rational Zero Theorem, Descartes' Rule of Signs, reduce to a quadratic, and then solve by using factoring or the Quadratic Formula.
f(x) = x4 + 6x3 + 7x2 - 6x - 8
Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website! One change in sign so expect one positive root, according to Descartes Rule of Signs.
(-x)^4+6(-x)^3+7(-x)^2-6(-x)-8
x^4-6x^3+7x^2+6x-8
That is three changes in sign, so expect three negative roots according to Rule of Signs.
Possible roots to check using Rational Roots Theorem and synthetic division would be -1, -2, -4, -8, 1, 2, 4, 8.
Best to check for the negative roots first because we expect MORE of them.
I will omit showing the actual synthetic divisions, but here is what I checked and what found:
Check -2: Yes, root. Zero remainder. Quotient  .
Check -4: Yes, root. Zero remainder. Quotient  .
And from that last quotient, we see that roots are +1 and -1.
Summary of Roots found:
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-4, -2, -1, and +1
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