How Do You Find the Quotient of a Polynomial?

To find the quotient of a polynomial division, you typically use polynomial long division, similar to the long division method used for numbers.

Steps for Polynomial Long Division:

1. Arrange: Ensure both the dividend (the polynomial being divided) and the divisor (the polynomial you're dividing by) are written in descending order of exponents. Add placeholders (terms with a coefficient of 0) for any missing exponents. For example, if you have x³ + 1 being divided by x + 1, write the dividend as x³ + 0x² + 0x + 1.

2. Divide: Divide the first term of the dividend by the first term of the divisor. This will be the first term of the quotient.

3. Multiply: Multiply the divisor by the term you just found in the quotient.

4. Subtract: Subtract the result from the corresponding terms in the dividend.

5. Bring Down: Bring down the next term from the dividend.

6. Repeat: Repeat steps 2-5 using the new polynomial you created until the degree of the remaining polynomial (the remainder) is less than the degree of the divisor.

7. Write the Result: The quotient is the polynomial you built in step 2. The remainder (if any) is the polynomial left after the final subtraction. Express the result as:

   Dividend / Divisor = Quotient + Remainder / Divisor

Example:

Let's divide (x² + 3x + 2) by (x + 1).

	x + 2
x + 1	x2 + 3x + 2
	x2 + x
-------	---------
	2x + 2
	2x + 2
-------	---------
	0

Explanation:

• Divide: x² / x = x (This is the first term of the quotient)
• Multiply: x * (x + 1) = x² + x
• Subtract: (x² + 3x + 2) - (x² + x) = 2x + 2
• Divide: 2x / x = 2 (This is the second term of the quotient)
• Multiply: 2 * (x + 1) = 2x + 2
• Subtract: (2x + 2) - (2x + 2) = 0 (Remainder)

Therefore, (x² + 3x + 2) / (x + 1) = x + 2 + 0/(x+1) = x + 2.

Alternative Methods

• Synthetic Division: A shorthand method for polynomial division, but it only works when the divisor is of the form (x - a), where a is a constant.

Key Considerations:

• Be mindful of signs during the subtraction steps.
• Always ensure polynomials are arranged in descending order of exponents.
• If a term is missing in the dividend, add a placeholder with a coefficient of 0.