Which of the following expressions is not a factor of x^5−x^4−40x^3+40x^2+144x−144
? Use long division to solve.
To determine if an expression is a factor of a polynomial, we can use long division.
Let's divide the polynomial x^5 - x^4 - 40x^3 + 40x^2 + 144x - 144 by each of the given expressions.
1) x - 2:
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x - 2 | x^5 - x^4 - 40x^3 + 40x^2 + 144x - 144
x^5 - 2x^4
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x^4 - 40x^3 + 40x^2
x^4 - 2x^3
______________
- 38x^3 + 40x^2 + 144x
- 38x^3 + 76x^2
______________
- 36x^2 + 144x - 144
- 36x^2 + 72x
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72x - 144
72x - 144
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0
Since the remainder is zero, x - 2 is a factor of the polynomial.
2) x + 2:
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x + 2 | x^5 - x^4 - 40x^3 + 40x^2 + 144x - 144
x^5 + 2x^4
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- 39x^4 - 40x^3
- 39x^4 - 78x^3
______________
38x^3 + 40x^2 + 144x
38x^3 + 76x^2
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- 36x^2 + 144x - 144
- 36x^2 - 72x
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216x - 144
216x + 432
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- 576
Since the remainder is not zero, x + 2 is not a factor of the polynomial.
Therefore, the expression x + 2 is not a factor of x^5 - x^4 - 40x^3 + 40x^2 + 144x - 144.
To determine which expression is not a factor of the given polynomial x^5 - x^4 - 40x^3 + 40x^2 + 144x - 144, we can use long division.
Let's divide x^5 - x^4 - 40x^3 + 40x^2 + 144x - 144 by each of the given expressions and see which one does not give a remainder of zero.
For the first expression, let's divide x - 8:
x^4 - 40x^2 + 8x + 144
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x - 8 | x^5 - x^4 - 40x^3 + 40x^2 + 144x - 144
- x^5 + 8x^4
___________
- 9x^4 - 40x^3
+ 9x^4 - 72x^3
_____________
- 112x^3 + 40x^2
+ 112x^3 - 896x^2
________________
- 856x^2 + 144x
+ 856x^2 - 6848x
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- 6704x + 144
+ 6704x - 5376
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- 5232
Since there is a remainder of -5232, x - 8 is not a factor of the polynomial.
Now let's divide the polynomial by each of the other expressions:
x + 3:
x^4 - 4x^3 - 52x^2 + 304x + 720
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x + 3 | x^5 - x^4 - 40x^3 + 40x^2 + 144x - 144
- x^5 - 3x^4
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+ 2x^4 - 40x^3
- 2x^4 - 6x^3
_____________
- 34x^3 + 40x^2
+ 34x^3 + 102x^2
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142x^2 + 144x
- 142x^2 - 426x
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- 282x - 144
+ 282x + 846
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702
Since there is a remainder of 702, x + 3 is also not a factor of the polynomial.
Finally, let's divide by x - 6:
x^4 + 2x^3 - 28x^2 + 288x + 864
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x - 6 | x^5 - x^4 - 40x^3 + 40x^2 + 144x - 144
- x^5 + 6x^4
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+ 7x^4 - 40x^3
- 7x^4 + 42x^3
_____________
+ 2x^3 + 40x^2
- 2x^3 + 12x^2
_____________
52x^2 + 144x
- 52x^2 + 312x
______________
456x - 144
- 456x + 2736
___________
2592
Since there is a remainder of 2592, x - 6 is also not a factor of the polynomial.
Therefore, the expression that is not a factor of the polynomial x^5 - x^4 - 40x^3 + 40x^2 + 144x - 144 is x - 8.
To find the expression that is not a factor of the given polynomial, we can use long division to divide the polynomial by each of the expressions and check the remainder. If the remainder is zero, then the expression is a factor. Otherwise, it is not a factor.
Let's start by performing long division with the first expression x^2 - x.
x^3 + 0x^2 - 39x + 144
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x^2 - x | x^5 - x^4 - 40x^3 + 40x^2 + 144x - 144
-(x^5 - x^4)
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0x^4 - 40x^3
40x^3 - 40x^2
_______________
0x^2 + 144x
-144x + 144
_______________
0
As we can see, there is no remainder, so x^2 - x is a factor of the given polynomial.
Now, let's perform long division with the second expression x^2 + 2x.
x^3 + x^2 - 42x + 144
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x^2 + 2x | x^5 - x^4 - 40x^3 + 40x^2 + 144x - 144
-(x^5 + 2x^4)
_______________________
-3x^4 - 40x^3
-3x^4 - 6x^3
___________________
0x^3 + 40x^2
40x^2 + 80x
______________________
64x - 144
64x + 128
______________
-272
As we can see, there is a remainder of -272, so x^2 + 2x is not a factor of the given polynomial.
Lastly, let's perform long division with the third expression x^2 - 64.
x^3 + x - 144
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x^2 - 64 | x^5 - x^4 - 40x^3 + 40x^2 + 144x - 144
-(x^5 - 64x^3)
__________________
63x^3 + 40x^2
63x^3 - 4032
______________
4044x^2 + 144x
4044x^2 - 258816
_______________
258960x - 144
258960x - 16527360
_________________
16527120
As we can see, there is a remainder of 16527120, so x^2 - 64 is not a factor of the given polynomial.
Therefore, the expression x^2 + 2x is not a factor of the polynomial x^5 - x^4 - 40x^3 + 40x^2 + 144x - 144.