detail the process of polynomial division when the divisor is a monomial. Demonstrate the process with an example. How does this process change when the divisor is not a monomial?
When performing polynomial division with a monomial divisor, we divide each term of the polynomial by the monomial divisor separately. Here is the step-by-step process:
1. Arrange the polynomial and the monomial divisor in descending order of exponents.
2. Start with the term having the highest power in the polynomial and divide it by the monomial divisor.
3. Write the result of this division as the first term of the quotient.
4. Multiply the quotient term by the monomial divisor.
5. Subtract the product obtained in step 4 from the original polynomial term.
6. Bring down the next term from the original polynomial and repeat steps 2 to 5 until all terms have been processed.
7. The final result is the quotient obtained from the division.
Let's demonstrate this process with an example. Consider the polynomial division:
(3x^3 + 2x^2 - 5x + 1) ÷ (x)
Step 1: Arrange in descending order: 3x^3 + 2x^2 - 5x + 1 ÷ x
Step 2: Divide the first term: (3x^3 ÷ x = 3x^2)
Step 3: Write the quotient: 3x^2
Step 4: Multiply the quotient by the divisor: (3x^2 * x = 3x^3)
Step 5: Subtract the product from the original term: (3x^3 + 2x^2 - 3x^3 = 2x^2)
Step 6: Bring down the next term: 2x^2 - 5x
Step 7: Repeat steps 2 to 5: (2x^2 - 5x) ÷ x, (2x^2 ÷ x = 2x)
Step 8: Write the quotient: 3x^2 + 2x
Step 9: Multiply the quotient by the divisor: (2x * x = 2x^2)
Step 10: Subtract the product: (2x^2 - 2x^2 = 0)
Step 11: Bring down the next term: 0 - 5x + 1
Step 12: Repeat steps 2 to 5: (-5x ÷ x = -5)
Step 13: Write the quotient: 3x^2 + 2x - 5
Step 14: Multiply the quotient by the divisor: (-5 * x = -5x)
Step 15: Subtract the product: (-5x - (-5x) = 0)
Step 16: No more terms, so the final quotient is: 3x^2 + 2x - 5
Now, if the divisor is not a monomial (e.g., (x + 2)), the process changes slightly. We use the long division method instead of dividing each term. Instead of dividing term by term, we divide the highest-degree terms and proceed from there. The process follows similar steps as explained above, but we focus on the leading terms.