Divide the polynomials using long division.
(y^5-1024)/(y-4)
Let's perform long division to divide the polynomials (y^5 - 1024) by (y - 4):
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y - 4 | y^5 + 0y^4 + 0y^3 + 0y^2 + 0y - 1024
We start by dividing the leading term y^5 by y, which gives us y^4. Multiply y^4 by (y - 4) to get y^5 - 4y^4. Subtract this from the original polynomial to get:
y^5 + 0y^4 + 0y^3 + 0y^2 + 0y - 1024
- (y^5 - 4y^4)
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4y^4 + 0y^3 + 0y^2 + 0y - 1024
Now, we bring down the next term, which is 0y^3 and divide 4y^4 by y. This gives us 4y^3. Multiply 4y^3 by (y - 4) to get 4y^4 - 16y^3. Subtract this from the current polynomial to get:
4y^4 + 0y^3 + 0y^2 + 0y - 1024
- (4y^4 - 16y^3)
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16y^3 + 0y^2 + 0y - 1024
We bring down the next term, which is 0y^2 and divide 16y^3 by y. This gives us 16y^2. Multiply 16y^2 by (y - 4) to get 16y^3 - 64y^2. Subtract this from the current polynomial to get:
16y^3 + 0y^2 + 0y - 1024
- (16y^3 - 64y^2)
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64y^2 + 0y - 1024
We bring down the next term, which is 0y, and divide 64y^2 by y. This gives us 64y. Multiply 64y by (y - 4) to get 64y^2 - 256y. Subtract this from the current polynomial to get:
64y^2 + 0y - 1024
- (64y^2 - 256y)
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256y - 1024
We bring down the last term, which is -1024 and divide 256y by y. This gives us 256. Multiply 256 by (y - 4) to get 256y - 1024. Subtract this from the current polynomial to get:
256y - 1024
- (256y - 1024)
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0
Since the result of this subtraction is 0, we have now fully divided (y^5 - 1024) by (y - 4) using long division.
Therefore, the quotient is y^4 + 4y^3 + 16y^2 + 64y + 256, and the remainder is 0.
Divide the polynomials using long division.
(b^4-256)/(b+4)
Let's perform long division to divide the polynomials (b^4 - 256) by (b + 4):
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b + 4 | b^4 + 0b^3 + 0b^2 + 0b - 256
We start by dividing the leading term b^4 by b, which gives us b^3. Multiply b^3 by (b + 4) to get b^4 + 4b^3. Subtract this from the original polynomial to get:
b^4 + 0b^3 + 0b^2 + 0b - 256
- (b^4 + 4b^3)
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-4b^3 + 0b^2 + 0b - 256
Now, we bring down the next term, which is 0b^2 and divide -4b^3 by b. This gives us -4b^2. Multiply -4b^2 by (b + 4) to get -4b^3 - 16b^2. Subtract this from the current polynomial to get:
-4b^3 + 0b^2 + 0b - 256
- (-4b^3 - 16b^2)
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16b^2 + 0b - 256
We bring down the next term, which is 0b, and divide 16b^2 by b. This gives us 16b. Multiply 16b by (b + 4) to get 16b^2 + 64b. Subtract this from the current polynomial to get:
16b^2 + 0b - 256
- (16b^2 + 64b)
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-64b - 256
We bring down the last term, which is -256, and divide -64b by b. This gives us -64. Multiply -64 by (b + 4) to get -64b - 256. Subtract this from the current polynomial to get:
-64b - 256
- (-64b - 256)
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0
Since the result of this subtraction is 0, we have now fully divided (b^4 - 256) by (b + 4) using long division.
Therefore, the quotient is b^3 - 4b^2 + 16b - 64, and the remainder is 0.
Divide the polynomials by using an appropriate method.
(2x^3-8x^2+3x-6)/(x-5)
To divide the polynomial (2x^3 - 8x^2 + 3x - 6) by (x - 5), we can use either long division or synthetic division. Let's use long division in this case.
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x - 5 | 2x^3 - 8x^2 + 3x - 6
We start by dividing the leading term 2x^3 by x, which gives us 2x^2. Multiply 2x^2 by (x - 5) to get 2x^3 - 10x^2. Subtract this from the original polynomial to get:
2x^3 - 8x^2 + 3x - 6
- (2x^3 - 10x^2)
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2x^2 + 3x - 6
Now, we bring down the next term, which is 3x, and divide 2x^2 by x. This gives us 2x. Multiply 2x by (x - 5) to get 2x^2 - 10x. Subtract this from the current polynomial to get:
2x^2 + 3x - 6
- (2x^2 - 10x)
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13x - 6
We bring down the last term, which is -6, and divide 13x by x. This gives us 13. Multiply 13 by (x - 5) to get 13x - 65. Subtract this from the current polynomial to get:
13x - 6
- (13x - 65)
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59
Since the result of this subtraction is 59, there is no remainder.
Therefore, the quotient is 2x^2 + 2x + 13, and the remainder is 59.