(x - 1)(x - 4); f(x) = x4 - 5x³ + 13x² - 45x + 36
We first find of
to show that the remainder is 0. That will show that (x - 1)
is a factor of f(x). I'll use synthetic division where we use +1
to divide by (x - 1):
1 | 1 -5 13 -45 36
| 1 -4 9 -36
1 -4 9 -36 0
So therefore (x - 1) is a factor, and we have now factored f(x) as
f(x) = (x - 1)(x³ - 4x² + 9x - 36)
Now we must also show that the factor in the second parentheses is
divisible by (x - 1)
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Now we must show that (x - 4) is also a fsctor
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We now find
to show that the remainder is 0. That will show that (x - 4)
is a factor of x³ - 4x² + 9x - 36, and therefore a factor of
f(x). I'll use synthetic division again this time where we
use +4 to divide by (x - 4):
4 | 1 -4 9 -36
| 4 0 36
1 0 9 0
So therefore (x - 4) is a factor, and we have now factored f(x) as
We started with"
f(x) = x4 - 5x³ + 13x² - 45x + 36
and factored it with synthetic division as this
f(x) = (x - 1)(x³ - 4x² + 9x - 36)
And further factored the right factor with synthetic
division as this:
f(x) = (x - 1)(x - 4)(x² + 0x + 9)
Eliminating the zero term, this is the factored form
f(x) = (x - 1)(x - 4)(x² + 9)
which proves that (x - 1)(x - 4) is a factor of f(x).
Edwin