The process of dividing one polynomial by another, resulting in a quotient and possibly a remainder.
Polynomial division is similar to long division with numbers, but it involves dividing terms with variables.
NoteThe degreeof a polynomial is the highest power of the variable in the polynomial.
Polynomial Division Process
Step 1: Arrange in Descending Order
Ensure both the dividend and divisor are arranged in descending order of their degrees.
Step 2: Divide the Leading Terms
- Divide the leading term of the dividend by the leading term of the divisor.
- Write the result as the first term of the quotient.
Step 3: Multiply and Subtract
- Multiply the entire divisor by the term obtained in Step 2.
- Subtract the result from the dividend.
Step 4: Repeat
- Repeat Steps 2 and 3 with the new polynomial obtained after subtraction.
- Continue until the degree of the remaining polynomial is less than the degree of the divisor.
Step 5: Remainder
The remaining polynomial is the remainder. If it is zero, the division is exact.
NoteEnsure to subtract carefully, as sign errors are common.
Checking the Result
To verify the division, multiply the quotient by the divisor and add the remainder. The result should be the original dividend.
Self reviewDivide \$3x^4 - 2x^3 + x - 5\$ by \$x^2 - 1\$. What is the quotient and remainder?
Theory of KnowledgeHow does the process of polynomial division relate to long division with numbers? What does this tell us about the nature of mathematical operations?
