Can we conclude that x - 4 is a factor of the polynomial

?

how do you know?

(big to answer correctly)

  No. We can conclude that (x-4) is not a factor of the polynomial.

Step-by-step explanation:

The polynomial remainder theorem tells you (x-4) will be a factor of the polynomial f(x) if and only if f(4) = 0. When we evaluate ...

  f(x) = 4x³ -20x² +18x -12

at x=4, we find that f(4) = -4. This is not zero, so (x-4) is not a factor of f(x).

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The evaluation is perhaps easiest done by hand when the function is written in Horner form:

  f(x) = ((4x -20)x +18)x -12

  f(4) = ((4·4 -10)4 +18)4 -12 = ((-4)4 +18)4 -12 = 2·4 -12

  f(4) = -4

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The one real factor of this polynomial is irrational, and the remaining factors are complex and irrational.