(2y^3 -3y^2 +3y -4)/(y-2)
To solve this problem, we can use polynomial division, also known as long division. Here's how you can do it step by step:
Step 1: Write the division problem in the long division format:
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y - 2 | 2y^3 - 3y^2 + 3y - 4
Step 2: Divide the first term of the dividend (2y^3) by the first term of the divisor (y). Place the result (2y^2) above the division bar and write it to the left of the dividend:
2y^2
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y - 2 | 2y^3 - 3y^2 + 3y - 4
Step 3: Multiply the divisor (y - 2) by the quotient (2y^2) and write the result below the dividend. Then, subtract this result from the dividend:
2y^2
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y - 2 | 2y^3 - 3y^2 + 3y - 4
- (2y^3 - 4y^2)
y^2 + 3y - 4
Step 4: Bring down the next term from the dividend (3y). The new dividend is now y^2 + 3y - 4:
2y^2 + (y + 2)
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y - 2 | 2y^3 - 3y^2 + 3y - 4
- (2y^3 - 4y^2)
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y^2 + 3y - 4
Step 5: Repeat steps 2 to 4 with the revised dividend (y^2 + 3y - 4) until there are no more terms left to bring down.
2y^2 + (y + 2) + (3)
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y - 2 | 2y^3 - 3y^2 + 3y - 4
- (2y^3 - 4y^2)
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y^2 + 3y - 4
- (y^2 - 2y)
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5y - 4
Step 6: Since there are no more terms left to bring down, the remainder is 5y - 4. Therefore, the expression (2y^3 - 3y^2 + 3y - 4)/(y-2) can be simplified as follows:
2y^2 + (y + 2) + (3) + (5y - 4)/(y - 2)
Hence, the simplified form of the expression is 2y^2 + y + 5 + (5y - 4)/(y - 2).