Suppose you divide a polynomial by a binomial. How do you know if the binomial is a factor of the polynomial? Create a sample problem that has a binomial which IS a factor of the polynomial being divided, and another problem that has a binomial which is NOT a factor of the polynomial being divided.
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one way is by using the factor theorem
if f(x) is divisible by (x-a) then f(a) = 0
an example would be x^3 - 2x^2 + 4x - 8
test to see if this is divisible by (x-2)
f(2) = 2^3 - 2(2)^2 + 4(2) - 8 = 0
so by factor theorem it is a factor
lets see if x+3 is a factor of the above polynomial
f(-3) = -3^3 -2(-3)^2 +4(-3) - 8 = -27 -18 - 12 - 8 = -65
so x+3 is not a factor
by another theorem (the remainder theorem) remainder = -65 for the division
The factor theorem is a special case of remainder theorem
Interesting, I see now. This is mine that I just came up with,
2x^4 - 9x^3 +21x^2 - 26x + 12 by 2x - 3.