Synthetic division is a shorthand method of dividing a polynomial by a linear factor of the form $x - k$. It's a simplified way to perform polynomial long division, especially useful when finding roots of polynomials or factoring them. The process involves using only the coefficients of the polynomial and the value 'k' to determine the quotient and remainder. This method is efficient and reduces the chance of errors compared to long division.
To perform synthetic division, write the coefficients of the polynomial horizontally, and place the value 'k' to the left. Bring down the first coefficient, multiply it by 'k', and write the result under the next coefficient. Add these two numbers together, and repeat the process until you reach the last coefficient. The last number is the remainder, and the other numbers are the coefficients of the quotient, which is one degree less than the original polynomial.
Match the term with its correct definition:
Synthetic division is a ______ method for dividing a polynomial by a ______ factor. It uses only the ______ of the polynomial and the value 'k'. The last number obtained is the ______, and the other numbers are the coefficients of the ______.
Explain in your own words why synthetic division is usually faster and more efficient than polynomial long division. Are there any situations where polynomial long division might be preferred?
Use synthetic division to solve the following problems. Show your work.
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