1) Find a special factoring pattern for the difference of two sextics: (a^6 - b^6)
2) Suppose that a cubic function f(x) = ax^3 + bx^2 + cx+ d can be factored into two terms, one of which is (x-n). Find the other tem using SYNTHETIC DIVISION and answer the following questions.
3) (ax^3+bx^2+cx+d)=(x-n) (______________)
4) What must be true about an^3+bn^2+cn+d ?
5) What does this tell us about value of f(n)
Any help with these questions is greatly appreciated, thanks.
+128731 1) a^6 - b^6 = factor as a difference of squares
(a^3 - b^3) (a^3 + b^3) = factor each as a difference of cubes
(a -b) (a^2 + ab + b^2) (a + b) (a^2 - ab + b^2)
+128731 2)
n [ a b c d ]
an an^2 + bn an^3 + bn^2 + cn
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a b+an an^2 + bn + c an^3 + bn^2 + cn + d
But if n is a factor, then an^3 + bn^2 + cn + d = 0
So
3) So .....ax^3 + bx^2 + cx + d = (x - n) * [ ax^2 + (b +an)x + (an^2 + bn + c) ]
4) As before, since n is a factor, an^3 + bn^2 + cn + d = 0
5) So f(n) = 0
