Essay: Show all work. Find the quotient: y^4 + 3y -5 / y^2 + 7
To find the quotient of two polynomials, you need to perform polynomial long division. Let's go step by step:
1. Write the dividend (the numerator) and the divisor (the denominator) in descending powers of the variable, y, with missing terms represented by placeholder zeros:
dividend: y^4 + 3y - 5
divisor: y^2 + 7
_______________________
y^2 + 7 | y^4 + 0y^3 + 0y^2 + 3y - 5
2. Divide the first term of the dividend by the first term of the divisor:
The first term of the dividend is y^4, and the first term of the divisor is y^2.
The quotient of y^4 divided by y^2 is y^2.
y^2
_________
y^2 + 7 | y^4 + 0y^3 + 0y^2 + 3y - 5
3. Multiply the divisor by the term we obtained (y^2) and write the result below the dividend with spaces left for each power of y:
y^2 * (y^2 + 7) = y^4 + 7y^2
y^2
_________
y^2 + 7 | y^4 + 0y^3 + 0y^2 + 3y - 5
- (y^4 + 7y^2)
__________________
-7y^2 + 3y - 5
4. Subtract the result of the multiplication from the original dividend:
(y^4 + 0y^3 + 0y^2 + 3y - 5) - (y^4 + 7y^2) = -7y^2 + 3y - 5
y^2
_________
y^2 + 7 | y^4 + 0y^3 + 0y^2 + 3y - 5
- (y^4 + 7y^2)
__________________
-7y^2 + 3y - 5
5. Bring down the next term from the original dividend, which is 3y:
The new dividend becomes -7y^2 + 3y - 5 + 3y = -7y^2 + 6y - 5
y^2 + (6y - 7)
_____________________
y^2 + 7 | y^4 + 0y^3 + 0y^2 + 3y - 5
- (y^4 + 7y^2)
___________________
-7y^2 + 3y - 5
- (-7y^2 + 6y - 5)
___________________
y - 0
6. Divide the first term of the new dividend (-7y^2) by the first term of the divisor (y^2):
The quotient of -7y^2 divided by y^2 is -7.
y^2 + (6y - 7)
_____________________
y^2 + 7 | y^4 + 0y^3 + 0y^2 + 3y - 5
- (y^4 + 7y^2)
___________________
-7y^2 + 3y - 5
- (-7y^2 + 6y - 5)
___________________
y - 0
7. Multiply the divisor by the term we obtained (-7) and write the result below the new dividend:
-7 * (y^2 + 7) = -7y^2 - 49
y^2 + (6y - 7)
_____________________
y^2 + 7 | y^4 + 0y^3 + 0y^2 + 3y - 5
- (y^4 + 7y^2)
___________________
-7y^2 + 3y - 5
- (-7y^2 + 6y - 5)
___________________
y - 0
-7y^2 - 49
8. Subtract the result of the multiplication from the new dividend:
(-7y^2 + 6y - 5) - (-7y^2 - 49) = 13y + 44
y^2 + (6y - 7) + (13y + 44)
_____________________________________
y^2 + 7 | y^4 + 0y^3 + 0y^2 + 3y - 5
- (y^4 + 7y^2)
___________________
-7y^2 + 3y - 5
- (-7y^2 + 6y - 5)
___________________
y - 0
-7y^2 - 49
- (13y + 44)
___________________
-14y - 49
9. Since the new dividend (-14y - 49) does not have any powers of y greater than y^2, we can stop the division.
Thus, the quotient is y^2 + (6y - 7) + (13y + 44) = y^2 + 19y + 37.
Therefore, the quotient of the polynomial division y^4 + 3y - 5 divided by y^2 + 7 is y^2 + 19y + 37.