6x^3+4x^2-2x+5 divided by 3x^2+2x+1
Using the long division method
To divide the polynomial 6x^3 + 4x^2 - 2x + 5 by 3x^2 + 2x + 1 using long division, follow these steps:
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3x^2 + 2x + 1 | 6x^3 + 4x^2 - 2x + 5
Step 1: Divide the highest degree term of the dividend (6x^3) by the highest degree term of the divisor (3x^2) to get 2x, which is then placed above the line.
2x
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3x^2 + 2x + 1 | 6x^3 + 4x^2 - 2x + 5
Step 2: Multiply the divisor (3x^2 + 2x + 1) by the quotient (2x) and subtract the result from the dividend (6x^3 + 4x^2 - 2x + 5).
2x (3x^2 + 2x + 1)
6x^3 + 4x^2 + 2x
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-6x^3 - 4x^2 + 2x + 5
Step 3: Bring down the next term from the dividend (-2x + 5) to continue the division process.
2x (3x^2 + 2x + 1)
6x^3 + 4x^2 + 2x
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-6x^3 - 4x^2 + 2x + 5
-2x + 5
Step 4: Repeat steps 1 and 2 with the new dividend (-6x^3 - 4x^2 + 2x + 5).
2x (3x^2 + 2x + 1)
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-6x^3 - 4x^2 + 2x + 5
-6x^3 - 4x^2 - 2x
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4x + 5
Step 5: Since the degree of the new dividend (4x + 5) is less than the degree of the divisor (3x^2 + 2x + 1), we have a remainder of 4x + 5.
Therefore, the quotient is 2x and the remainder is 4x + 5.
To divide 6x^3 + 4x^2 - 2x + 5 by 3x^2 + 2x + 1 using long division, follow these steps:
Step 1: Determine the highest power of x in the dividend and the divisor, and write them in descending order:
Dividend: 6x^3 + 4x^2 - 2x + 5
Divisor: 3x^2 + 2x + 1
Step 2: Divide the terms with the highest power. The result will give the first term of the quotient:
6x^3 ÷ 3x^2 = 2x
Step 3: Multiply the entire divisor by the first term of the quotient:
2x * (3x^2 + 2x + 1) = 6x^3 + 4x^2 + 2x
Step 4: Subtract the product obtained in step 3 from the dividend, then bring down the next term:
(6x^3 + 4x^2 - 2x + 5) - (6x^3 + 4x^2 + 2x) = -4x + 5
Step 5: Repeat steps 2-4 with the updated dividend (-4x + 5):
Dividend: -4x + 5
Divisor: 3x^2 + 2x + 1
-4x ÷ 3x^2 = 0
0 * (3x^2 + 2x + 1) = 0
(-4x + 5) - 0 = -4x + 5
Step 6: The result in step 5 is the second term of the quotient:
-4x ÷ 3x^2 = 0
Step 7: Repeat steps 2-4 with the updated dividend (-4x + 5):
Dividend: -4x + 5
Divisor: 3x^2 + 2x + 1
-4x ÷ 3x^2 = 0
0 * (3x^2 + 2x + 1) = 0
(-4x + 5) - 0 = -4x + 5
Step 8: Since the resulting dividend (-4x + 5) has a smaller degree than the divisor (3x^2 + 2x + 1), we cannot divide further.
Therefore, the quotient is 2x + 0, or simply 2x, and the remainder is -4x + 5.
To divide 6x^3 + 4x^2 - 2x + 5 by 3x^2 + 2x + 1 using the long division method, follow these steps:
1. Start by comparing the highest power of x in the dividend (6x^3) with the highest power of x in the divisor (3x^2). Divide 6x^3 by 3x^2 to get 2x.
2. Multiply the entire divisor (3x^2 + 2x + 1) by the quotient obtained (2x). The result is 6x^3 + 4x^2 + 2x.
3. Subtract the result (6x^3 + 4x^2 + 2x) from the dividend (6x^3 + 4x^2 - 2x + 5). This gives you -4x - 2x + 5.
4. Bring down the next term from the dividend, which is -4x. Now you have -4x - 2x + 5.
5. Compare the highest power of x in the new expression (-4x) with the highest power of x in the divisor (3x^2). Divide -4x by 3x^2 to obtain -4/3x.
6. Multiply the entire divisor by the new quotient (-4/3x). This gives you -4x - 8/3x - 4/3.
7. Subtract the result (-4x - 8/3x - 4/3) from the previous expression (-4x - 2x + 5). This results in 2/3x + 17/3.
8. Bring down the next term, which is 2/3x. Now the expression becomes 2/3x + 17/3.
9. Compare the highest power of x in the new expression (2/3x) with the highest power of x in the divisor (3x^2). Since the divisor's power is greater, there are no more terms to bring down.
10. Write the final remainder (2/3x + 17/3) over the divisor (3x^2 + 2x + 1) to get the quotient.
So, the quotient is 2x - 4/3 + (2/3x + 17/3) / (3x^2 + 2x + 1).