Divide the polynomials using long division.
(x^3-28x^2-21x+20)/(x+4)
To divide the polynomials using long division, we start by writing the equation in the long division format:
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(x+4) | x^3 - 28x^2 - 21x + 20
We will focus on dividing the highest degree term of the numerator, which is x^3, by the highest degree term of the denominator, which is x. The result of this division is x^2, so we write x^2 above the line:
x^2
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(x+4) | x^3 - 28x^2 - 21x + 20
Next, we multiply the entire denominator (x+4) by the x^2 we just found, and write the result below the line, lining up like terms:
x^2
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(x+4) | x^3 - 28x^2 - 21x + 20
- x^3 - 4x^2
Next, we subtract the new polynomial (x^3 + 4x^2) from the original polynomial (x^3 - 28x^2 - 21x + 20):
x^2
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(x+4) | x^3 - 28x^2 - 21x + 20
- x^3 - 4x^2
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-24x^2 - 21x + 20
Now, we repeat the process with the new polynomial (-24x^2 - 21x + 20). We focus on dividing the highest degree term (-24x^2) by the highest degree term (x) in the denominator. The result is -24x, so we write -24x below the line, below the x^2 we already found:
x^2 - 24x
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(x+4) | x^3 - 28x^2 - 21x + 20
- x^3 - 4x^2
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-24x^2 - 21x + 20
Again, we multiply the entire denominator (x+4) by the -24x, and write the result below the line, lining up like terms:
x^2 - 24x
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(x+4) | x^3 - 28x^2 - 21x + 20
- x^3 - 4x^2
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-24x^2 - 21x + 20
+ 24x^2 + 96x
Next, we subtract the new polynomial (24x^2 + 96x) from the previous polynomial (-24x^2 - 21x + 20):
x^2 - 24x
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(x+4) | x^3 - 28x^2 - 21x + 20
- x^3 - 4x^2
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-24x^2 - 21x + 20
+ 24x^2 + 96x
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75x + 20
Now, we repeat the process with the new polynomial (75x + 20). We focus on dividing the highest degree term (75x) by the highest degree term (x) in the denominator. The result is 75, so we write 75 below the line, below the x term we already found:
x^2 - 24x + 75
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(x+4) | x^3 - 28x^2 - 21x + 20
- x^3 - 4x^2
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-24x^2 - 21x + 20
+ 24x^2 + 96x
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75x + 20
Finally, we multiply the entire denominator (x+4) by the 75, and write the result below the line, lining up like terms:
x^2 - 24x + 75
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(x+4) | x^3 - 28x^2 - 21x + 20
- x^3 - 4x^2
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-24x^2 - 21x + 20
+ 24x^2 + 96x
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75x + 20
- 75x - 300
Now, we subtract the new polynomial (-75x - 300) from the previous polynomial (75x + 20):
x^2 - 24x + 75
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(x+4) | x^3 - 28x^2 - 21x + 20
- x^3 - 4x^2
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-24x^2 - 21x + 20
+ 24x^2 + 96x
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75x + 20
- 75x - 300
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320
At this point, we have reached a remainder of 320 with no more terms left in the numerator to divide by the denominator, so our final answer is:
x^2 - 24x + 75 + (320 / (x + 4))
whats that answer?
The final answer is:
x^2 - 24x + 75 + 320/(x + 4)
To divide the given polynomials using long division, follow these steps:
Step 1: Arrange the polynomial in descending order of powers.
(x^3 - 28x^2 - 21x + 20) ÷ (x + 4)
Step 2: Begin the long division process. Divide the highest power term of the dividend by the highest power term of the divisor.
x^3 ÷ x = x^2
Step 3: Multiply the entire divisor (x + 4) by the quotient obtained in the previous step (x^2).
(x^2) * (x + 4) = x^3 + 4x^2
Step 4: Subtract the product obtained in Step 3 from the original dividend.
(x^3 - 28x^2 - 21x + 20) - (x^3 + 4x^2) = -32x^2 - 21x + 20
Step 5: Bring down the next term from the original dividend, which is -32x^2.
-32x^2
Step 6: Repeat the division process by dividing the current term (-32x^2) by the divisor (x + 4).
-32x^2 ÷ x = -32x
Step 7: Multiply the entire divisor (x + 4) by the new quotient (-32x).
(-32x) * (x + 4) = -32x^2 - 128x
Step 8: Subtract the product obtained in Step 7 from the remaining polynomial obtained in Step 4.
(-32x^2 - 21x + 20) - (-32x^2 - 128x) = 107x + 20
Step 9: Bring down the next term from the original dividend, which is 107x.
107x
Step 10: Repeat the division process by dividing the current term (107x) by the divisor (x + 4).
107x ÷ x = 107
Step 11: Multiply the entire divisor (x + 4) by the new quotient (107).
(107) * (x + 4) = 107x + 428
Step 12: Subtract the product obtained in Step 11 from the remaining polynomial obtained in Step 8.
(107x + 20) - (107x + 428) = -408
Step 13: At this point, we have no more terms to bring down. The division process is complete.
Step 14: The quotient is obtained by combining the quotients obtained in Steps 2, 6, and 10.
Quotient = x^2 - 32x + 107
Step 15: The remainder is the constant term obtained in Step 12.
Remainder = -408
Therefore, the answer to the division is:
(x^3 - 28x^2 - 21x + 20) ÷ (x + 4) = x^2 - 32x + 107 with a remainder of -408.