Question 72593: I need someone really smart.
Let f(x) =x3 -8x2 +17x -9. use the factor theorem to find other solutions to f(x) -f(1) = 0, beside x=1.
I have 6 algebra books that can not explain this factor theorem whereas I can understand it, Can someone out there explain an easy way to understand this.
Thank you
Answer by rmromero(383)   (Show Source):
You can put this solution on YOUR website!
Let f(x) =x3 -8x2 +17x -9. use the factor theorem to find other solutions to f(x) -f(1) = 0, beside x=1.
First let us evaluate f(x) =x3 -8x2 +17x -9, given f(1)
Using Factor theorem:
f(1) = x3 -8x2 +17x -9
= 1^3 - 8(1)^2 + 17(1) - 9
= 1 - 8 + 17 - 9
= 18 - 17
f(1) = 1
Condition: f(x) -f(1) = 0 ,
1 - 1 = 0 right?
When f(x) = x3 -8x2 +17x -9 , f(x) = 1, f(x) -f(1) = 0
f(x) = x3 -8x2 +17x -9
1 = x3 -8x2 +17x -9
0 = x3 -8x2 +17x -10 <<<<<< Find roots of the polynomial
Any factor of f(x) = x3 -8x2 +17x -10 will make f(x) -f(1) = 0.Let us find
possible factors/roots of f(x) = x3 -8x2 +17x -10.
Possible roots = Factor of the constant 10 over coefficient x^3 which is 1.
Factors of 10 = 1, -1, 2, -2, 5, -5, 10, -10
Factors of 1 = 1, -1
Possible roots are: {1, -1, 2, -2, 5, -5, 10, -10}
Use Factor theorem to test which of the following are
the root of the polynomial
f(1) = 1^3 -8(1)^2 +17(1) -10
= 1 - 8 + 17 - 10
= 18 - 18
= 0 ----------> This is one root of the polynomial
f(-1) = (-1)^3 -8(-1)^2 +17(-1) -10
= -1 - 8 - 17 - 10
= -36 ---------> since this has remainder, this is not a root of the polynomial
F(2) = 2^3 -8(2)^2 +17(2) -10
= 8 - 32 + 34 - 10
= 42 - 42
= 0 --------> Root of the polynomial
f(-2) = (-2)^3 -8(-2)^2 +17(-2) -10
= -8 - 32 - 34 - 10
= -84 ----------> not a root
f(5) = 5^3 -8(5)^2 +17(5) -10
= 125 - 200 + 85 - 10
= 210 - 210
= 0 ----------> root of the polynomial
f(-5) = (-5)^3 -8(-5)^2 +17(-5) -10
= -125 - 200 - 85 - 10
= - 420 ----> not a root
f(10) = 10^3 -8(10)^2 +17(10) -10
= 1000 - 800 + 170 - 10
= 1170 - 810
= 360 ----------> not a root
f(-10) = (-10)^3 -8(-10)^2 +17(-10) -10
= -1000 - 800 - 170 - 10
= -1980 -------------> not a root
Therefore aside from x = 1, the other solutions are x = 2 and x = 5
Checking:
f(x) =x3 -8x2 +17x -9, f(x) - f(1) = 0
when x = 2
f(2) - f(1) = 0
(2^3 -8(2)^2 +17(2)-9) - (1^3 -8(1)^2 +17(1) -9)= 0
8 - 32 + 34 - 9 - (1 - 8 + 17 - 9) = 0
42 - 41 -1 = 0
0 = 0
When x = 5
f(5) - f(1) = 0
5^3 -8(5)^2 +17(5)- 9 - (1^3 -8(1)^2 +17(1) -9) = 0
125 - 200 + 85 - 9 - (1 - 8 + 17 - 9) = 0
210 - 209 - 1 = 0
0 = 0
If you have questions, just ask ok.
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