Dividing Polynomials
Overview
Dividing polynomials involves dividing a polynomial by either a monomial or another polynomial. There are two primary methods for dividing polynomials:
- Long Division: This method is used when dividing a polynomial by another polynomial of degree 1 or higher.
- Synthetic Division: This is a shortcut method for dividing a polynomial by a binomial of the form \( (x - c) \).
Steps for Polynomial Long Division
- Divide the first term of the dividend by the first term of the divisor.
- Multiply the entire divisor by this result.
- Subtract the result from the dividend to get the new remainder.
- Repeat the process with the new remainder until you get a remainder of degree less than the divisor.
Example 1: Long Division
Divide \( 2x^3 + 3x^2 - 5x + 6 \) by \( x + 2 \):
Step 1: Divide \( 2x^3 \) by \( x \) to get \( 2x^2 \).
Step 2: Multiply \( (x + 2) \) by \( 2x^2 \) to get \( 2x^3 + 4x^2 \).
Step 3: Subtract \( (2x^3 + 4x^2) \) from \( (2x^3 + 3x^2 - 5x + 6) \) to get \( -x^2 - 5x + 6 \).
Step 4: Divide \( -x^2 \) by \( x \) to get \( -x \).
Step 5: Multiply \( (x + 2) \) by \( -x \) to get \( -x^2 - 2x \).
Step 6: Subtract \( (-x^2 - 2x) \) from \( (-x^2 - 5x + 6) \) to get \( -3x + 6 \).
Step 7: Divide \( -3x \) by \( x \) to get \( -3 \).
Step 8: Multiply \( (x + 2) \) by \( -3 \) to get \( -3x - 6 \).
Step 9: Subtract \( (-3x - 6) \) from \( (-3x + 6) \) to get a remainder of 12.
The result is: \( \frac{2x^2 - x - 3}{x + 2} + \frac{12}{x + 2} \).
Example 2: Synthetic Division
Divide \( x^3 + 4x^2 - 2x + 1 \) by \( x - 1 \):
Step 1: Set up the synthetic division table with the coefficients of the dividend and the root of the divisor (\( x - 1 \), so root is 1):
1 | 1 4 -2 1
| 1 5 3
-------------------
1 5 3 4
Step 2: The quotient is \( x^2 + 5x + 3 \) with a remainder of 4.
Practice Questions
- Question 1: Divide the following polynomial using long division:
\( 4x^3 + 2x^2 - 6x + 8 \div 2x + 4 \)
Solution
Step 1: Divide \( 4x^3 \) by \( 2x \) to get \( 2x^2 \).
Step 2: Multiply \( (2x + 4) \) by \( 2x^2 \) to get \( 4x^3 + 8x^2 \).
Step 3: Subtract \( (4x^3 + 8x^2) \) from \( (4x^3 + 2x^2 - 6x + 8) \) to get \( -6x^2 - 6x + 8 \).
Step 4: Divide \( -6x^2 \) by \( 2x \) to get \( -3x \).
Step 5: Multiply \( (2x + 4) \) by \( -3x \) to get \( -6x^2 - 12x \).
Step 6: Subtract \( (-6x^2 - 12x) \) from \( (-6x^2 - 6x + 8) \) to get \( 6x + 8 \).
Step 7: Divide \( 6x \) by \( 2x \) to get \( 3 \).
Step 8: Multiply \( (2x + 4) \) by \( 3 \) to get \( 6x + 12 \).
Step 9: Subtract \( (6x + 12) \) from \( (6x + 8) \) to get a remainder of -4.
The result is: \( \frac{2x^2 - 3x + 3}{2x + 4} - \frac{4}{2x + 4} \).
- Question 2: Divide the following polynomial using synthetic division:
\( x^4 + 3x^3 - 2x^2 - 5x + 4 \div x + 2 \)
Solution
Set up the synthetic division table with the root \( -2 \):
-2 | 1 3 -2 -5 4
| -2 -2 8 -6
---------------------
1 1 -4 3 -2
The quotient is \( x^3 + x^2 - 4x + 3 \) with a remainder of -2.
- Question 3: Divide the following polynomial using long division:
\( 3x^4 + 2x^3 - 7x^2 + 5x - 6 \div x^2 - x \)
Solution
Follow the steps of long division:
Step 1: Divide \( 3x^4 \) by \( x^2 \) to get \( 3x^2 \).
Step 2: Multiply \( (x^2 - x) \) by \( 3x^2 \) to get \( 3x^4 - 3x^3 \).
Step 3: Subtract \( (3x^4 - 3x^3) \) from \( (3x^4 + 2x^3 - 7x^2 + 5x - 6) \) to get \( 5x^3 - 7x^2 + 5x - 6 \).
Step 4: Divide \( 5x^3 \) by \( x^2 \) to get \( 5x \).
Step 5: Multiply \( (x^2 - x) \) by \( 5x \) to get \( 5x^3 - 5x^2 \).
Step 6: Subtract \( (5x^3 - 5x^2) \) from \( (5x^3 - 7x^2 + 5x - 6) \) to get \( -2x^2 + 5x - 6 \).
Step 7: Divide \( -2x^2 \) by \( x^2 \) to get \( -2 \).
Step 8: Multiply \( (x^2 - x) \) by \( -2 \) to get \( -2x^2 + 2x \).
Step 9: Subtract \( (-2x^2 + 2x) \) from \( (-2x^2 + 5x - 6) \) to get \( 3x - 6 \).
The result is: \( 3x^2 + 5x - 2 + \frac{3x - 6}{x^2 - x} \).