Question

use the factor theorem and synthetic division to decide whether or not x-4 is a factor o f x^3-2x^2+x-1.

Answer 1

plug in 4 to see if the function evaluate to zero

Answer 2

synthetic division. write the coefficients

1 -2 1 -1

Answer 3

put a 4 on the side because you are dividing by x - 4

1 -2 1 -1

4

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Answer 4

bring down the 1

1 -2 1 -1


4
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1

Answer 5

4 times 1 is 4

1 -2 1 -1


4 4
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1

Answer 6

-2+4=2
1 -2 1 -1


4 4
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1 2

Answer 7

4*2=8

1 -2 1 -1


4 4 8
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1 2

Answer 8

8+1=9

1 -2 1 -1


4 4 8
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1 2 9

Answer 9

4*9=36


1 -2 1 -1


4 4 8 36
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1 2 9

Answer 10

-1+36=36
1 -2 1 -1


4 4 8 36
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1 2 9 35

Answer 11

so the answer is no, it does not divide evenly. it leaves a remainder of 35

Answer 12

and in fact you now know that
\[\frac{x^3-2x^2+x-1}{x-4}=x^2+2x+9+\frac{35}{x-4}\]

Answer 13

or to put it another way,
\[x^3-2x^2+x-1=(x-4)(x^2-2x+9) +35

Answer 14

\[x^3-2x^2+x-1=(x-4)(x^2-2x+9) +35\]