Dividing polynomial functions involves a process similar to long division with numbers, where you divide the dividend (the polynomial being divided) by the divisor (the polynomial doing the dividing) to find the quotient and potentially a remainder. Here’s a step-by-step guide on how to divide polynomial functions:
Long Division Method for Polynomials
This method is most commonly used and is very similar to numerical long division. Here's how it works, incorporating information from the reference provided:
- Set Up the Division Problem:
- Write the dividend inside the long division symbol and the divisor outside. Ensure both polynomials are written in descending order of their exponents, and include 0 coefficients for missing terms.
- Divide the Leading Terms:
- Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor. This gives the first term of the quotient which will sit above the dividend inside the division bar.
- Multiply:
- Multiply the first term of the quotient by the entire divisor and place the result below the dividend, aligning like terms.
- Subtract:
- Subtract the result obtained in step 3 from the dividend. This might mean adding the inverse if that is easier. Be sure to change the sign of each term when subtracting.
- Bring Down:
- Bring down the next term of the dividend.
- Repeat:
- Repeat steps 2-5 using the new polynomial as the dividend, until there are no further terms to bring down from the original dividend or the degree of the remaining polynomial is less than the degree of the divisor.
Example
Let's illustrate with a simple example: Divide (x^3 + 6x^2 + 11x + 6) by (x+1).
| x² | + 5x | + 6 | ||
|---|---|---|---|---|
| x + 1 | x³ + 6x² + 11x + 6 | |||
| -(x³+x²) | ||||
| 5x² + 11x | ||||
| -(5x²+5x) | ||||
| 6x + 6 | ||||
| -(6x+6) | ||||
| 0 |
Thus, (x^3 + 6x^2 + 11x + 6) / (x+1) = x^2 + 5x + 6.
Key Points to Remember
- Placeholders: Include 0 coefficients for any missing terms in either the dividend or divisor, like when there is no x² term in a polynomial with x³. This is important for aligning terms during the division process.
- Remainder: If, after the final subtraction, there is a non-zero polynomial with a lower degree than the divisor, this polynomial is the remainder. Write the remainder as a fraction with the divisor as the denominator, and add it to the quotient.
- Verification: You can check your answer by multiplying the quotient by the divisor, then adding the remainder (if any). The result should equal the original dividend.
- Alternative Methods: Synthetic division can be used as a faster method, but it only works when dividing by a linear divisor of the form x - a.
Conclusion
Dividing polynomial functions using long division follows a methodical procedure akin to numerical long division, involving repeated steps of dividing leading terms, multiplying, subtracting, and bringing down terms. By methodically completing each step, you can determine the quotient and remainder of any two polynomial functions.
