Essay. Show all work. find the quotient: y^4+3y-1
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y^2-3
To find the quotient of the given expression, you can use long division. Here is the step-by-step process:
Step 1: Divide the leading term of the dividend (y^4) by the leading term of the divisor (y^2) to get y^2. Write this as the first term of the quotient.
Step 2: Multiply the divisor (y^2-3) by the first term of the quotient (y^2). This gives you y^4 - 3y^2.
Step 3: Subtract the result obtained in Step 2 from the dividend (y^4 + 3y - 1). This can be done by changing the sign of each term and then combining like terms.
(y^4 + 3y - 1) - (y^4 - 3y^2) = (y^4 - y^4) + (3y + 3y^2) - 1 = 3y^2 + 3y - 1.
Step 4: Repeat Steps 1, 2, and 3 with the new expression obtained in Step 3, which is 3y^2 + 3y - 1.
Step 5: Divide the leading term of the new expression (3y^2) by the leading term of the divisor (y^2) to get 3. Write this as the next term of the quotient.
Step 6: Multiply the divisor (y^2-3) by the new term of the quotient (3). This gives you 3y^2 - 9.
Step 7: Subtract the result obtained in Step 6 from the new expression (3y^2 + 3y - 1). Again, change the sign of each term and combine like terms.
(3y^2 + 3y - 1) - (3y^2 - 9) = (3y^2 - 3y^2) + (3y - 3y) - (-9 - 1) = 6y + 8.
Step 8: Repeat Steps 1, 2, and 3 with the new expression obtained in Step 7, which is 6y + 8.
Step 9: Divide the leading term of the new expression (6y) by the leading term of the divisor (y^2) to get 0. Write this as the next term of the quotient.
Step 10: Multiply the divisor (y^2-3) by the new term of the quotient (0). This gives you 0.
Step 11: Subtract the result obtained in Step 10 from the new expression (6y + 8).
(6y + 8) - 0 = 6y + 8.
Step 12: Since the result obtained in Step 11 is a polynomial of degree 1 (6y), and the degree of the divisor (y^2-3) is 2, the division process is complete.
The final quotient of the given expression (y^4 + 3y - 1) divided by (y^2-3) is:
y^2 + 3 + (6y + 8) / (y^2 - 3).