Remaining theorem of polynomial if divided by x-k
Suppose the value of the polynomial f(k) = 0, then the
polynomial f(x) remains 0 if divided by x - k. Thus, (x - k) is said to be a
factor of f(x).
Polynomial factor theorem
The polynomial f(x) has a factor (x - k) if and only
if f(k) = 0
Example 1.
Show that x - 3 is a factor of the polynomial f(x) = x3 – x2 – x – 15
Answers:
Apparently, the result is f(x) = 0, so x - 3 is a factor of f(x) or x3 –
x2 – x – 15 = (x -3)(x2 + 2x + 5).
Example 2.
Solve the polynomial equation x3 – 11x2 + 30x – 8 by factoring on rational factors.
Answer:
For example, one of these factors is x - k, then k is a factor of (- 8). Possible factors of (- 8) are ± 1, ± 2, ± 4 and ± 8.
By experimenting with some of the numbers above, we find the remainder of the division 0 for x = 4, that is
So, the rational factor is x - 4.
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