In your own words, 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?
How can we give the answer "in your own words"?
Polynomial division is a method used to divide one polynomial by another polynomial. When the divisor is a monomial, the process is comparatively simpler. Let's go through the steps and also provide an example to make things clearer.
When dividing a polynomial by a monomial, we divide each term of the polynomial by the monomial. The goal is to divide the leading term of the polynomial by the monomial, and then subtract the result from the original polynomial. This process is repeated until all terms have been divided.
Here's an example to help illustrate the process:
Divide 3x^3 - 6x^2 + 9x + 12 by 3x.
Step 1: Start with the highest degree term.
Divide 3x^3 by 3x, which gives x^2. Write this as the first term of the quotient.
Step 2: Multiply the divisor (3x) by the quotient's first term (x^2).
Multiply 3x by x^2, which gives 3x^3. Write this below the original polynomial.
Step 3: Subtract the result from the original polynomial.
Subtract 3x^3 from 3x^3 - 6x^2 + 9x + 12. This gives - 6x^2 + 9x + 12.
Step 4: Bring down the next term.
Bring down the next term, which is -6x^2. Write it next to the previous result.
Step 5: Divide the new polynomial by the divisor.
Divide -6x^2 by 3x, which gives -2x. Write this as the second term of the quotient.
Step 6: Multiply the divisor (3x) by the second term of the quotient (-2x).
Multiply 3x by -2x, which gives -6x^2. Write this below the previous result.
Step 7: Subtract the result from the new polynomial.
Subtract -6x^2 from -6x^2 + 9x + 12. This gives 9x + 12.
Step 8: Bring down the next term.
Bring down the next term, which is 9x. Write it next to the previous result.
Step 9: Divide the new polynomial by the divisor.
Divide 9x by 3x, which gives 3. Write this as the third term of the quotient.
Step 10: Multiply the divisor (3x) by the third term of the quotient (3).
Multiply 3x by 3, which gives 9x. Write this below the previous result.
Step 11: Subtract the result from the new polynomial.
Subtract 9x from 9x + 12. This gives 12.
Step 12: Bring down the next term.
Bring down the next term, which is 12. Write it next to the previous result.
Since there are no more terms, we have completed the division.
The quotient is x^2 - 2x + 3, and the remainder is 12.
When the divisor is not a monomial, the process is more complex. In that case, we use long division or synthetic division to divide the polynomial. The steps involve comparing the leading term of the divisor with the leading term of the polynomial, dividing, and then multiplying the result back to the divisor. This process continues until the polynomial is fully divided or the degree of the remaining polynomial is less than the degree of the divisor.