Given f(x) = x^3 + 8x^2 + 17x + 10
We know that (x+5) and (x+1) are factors.
First, we divide f(x) by (x+5) using long division or synthetic division:
(x^3 + 8x^2 + 17x + 10) ÷ (x + 5)
(x + 5)|(x^3 + 8x^2 + 17x + 10)
- (x^3 + 5x^2)
-------------------
3x^2 + 17x
- (3x^2 + 15x)
----------------
2x + 10
- (2x + 10)
------------
0
Since there is no remainder, (x+5) is indeed a factor of f(x). Now we divide the resulting expression (x^2 + 3x + 2) by (x+1) to find the remaining factor:
(x^2 + 3x + 2) ÷ (x + 1)
(x + 1)|(x^2 + 3x + 2)
- (x^2 + x)
-------------------
2x + 2
- (2x + 2)
----------
0
Therefore, the remaining factor is x + 2.
The factored form of f(x) is: f(x) = (x + 5)(x + 1)(x + 2)
Using the factors (x+5) and (x+1) find the remaining factor of f(x)=x^3+8x^2+17x+10
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