Divide and Check
4x^4y-8x^6y^2+12x^8y^6
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4x^4y
When dividing fractions, we divide the
coefficients and subtract the exponents in the denominator from those
in the numerator.
4X^4y/4X^4y - 8X^6y^2/4X^4y + 12X^8y^6/4X^4y =
1 - 2X^(6y^2-4y) + 3X^(8y^6-4y)
Factor the exponents of each term:
1 - 2X^2y(3y-2) + 3X^4y(2y^5-1).
To divide the expression (4x^4y - 8x^6y^2 + 12x^8y^6) by 4x^4y, we will use polynomial long division. Polynomial long division is similar to long division with numbers, but with variables and exponents.
Step 1: First, arrange the expression and the divisor in descending order of the exponents of x.
The expression: 12x^8y^6 - 8x^6y^2 + 4x^4y
The divisor: 4x^4y
Step 2: Divide the first term of the expression (12x^8y^6) by the first term of the divisor (4x^4y). The result is 3x^4y^4.
Step 3: Multiply the entire divisor (4x^4y) by the result obtained in Step 2 (3x^4y^4). The result is 12x^8y^6.
Step 4: Subtract the result obtained in Step 3 from the original expression. Ignore any terms that cancel out.
Expression - 12x^8y^6 = -8x^6y^2 + 4x^4y
Step 5: Bring down the next term from the original expression, which is -8x^6y^2.
The current expression: -8x^6y^2 + 4x^4y
Step 6: Divide the first term of the current expression (-8x^6y^2) by the first term of the divisor (4x^4y). The result is -2x^2y^2.
Step 7: Multiply the entire divisor (4x^4y) by the result obtained in Step 6 (-2x^2y^2). The result is -8x^6y^2.
Step 8: Subtract the result obtained in Step 7 from the current expression. Ignore any terms that cancel out.
Current expression - (-8x^6y^2) = 4x^4y
Step 9: Bring down the next term from the current expression, which is 4x^4y.
The current expression: 4x^4y
As there are no more terms to bring down, we have finished dividing.
The quotient is 3x^4y^4 - 2x^2y^2 and the remainder is 0.