use synthetic division to show that x is a solution of the third-degree polynomial equation and use the result to factor the polynomial completely. list all the real zeros of the function
x3-7x+6=0, x=2
+6251 \(\mbox{If }x=2 \mbox{ is a root of }x^3-7x+6 \mbox{ then } (x-2) \mbox{ divides } x^3-7x+6\)
\(x^3-7x+6 = x^2(x-2) + (2x^2-7x+6) \\ (2x^2-7x+6)=2x(x-2) +(-3x + 6) \\ (-3x+6) = -3(x-2) \\ x^3-7x+6 = (x-2)(x^2+2x-3)\)
\(\mbox{To find the rest of the zeros we have to factor }x^2+2x-3 \\ \mbox{and this is easily done resulting in } \\ x^2+2x-3 = (x+3)(x-1) \\ \mbox{and thus the other zeros are } x=-3, x=1\)
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