Which method is most efficient for dividing a polynomial by a binomial of the form x - c?

Which method is most efficient for dividing a polynomial by a binomial of the form x - c?

• A. Polynomial Long Division
• B. Synthetic Division
• C. Rational Zero Theorem
• D. Quadratic Formula

Answer: (B)

Dividing a polynomial by a binomial of the form x - c is most efficiently done using synthetic division. This method takes advantage of the fact that the divisor has a simple root at x = c, so you can work directly with the coefficients rather than writing out full x-terms. In synthetic division, you lay out the coefficients of the polynomial and bring down the leading coefficient as the first coefficient of the quotient. Then you repeatedly multiply the current quotient coefficient by c and add that product to the next coefficient. Repeat across all coefficients. The numbers you get in the process give you the quotient, and the final sum is the remainder. This single-pass procedure uses only the coefficients and the constant c, making it faster and with less writing than ordinary polynomial long division. Other methods don’t fit this task as neatly. Polynomial long division can do the job, but it involves more algebraic clutter with x-terms and powers, so it’s slower. The Rational Zero Theorem helps identify possible roots, not perform division. The Quadratic Formula solves equations of degree two, not general division by a binomial like x - c.

Dividing a polynomial by a binomial of the form x - c is most efficiently done using synthetic division. This method takes advantage of the fact that the divisor has a simple root at x = c, so you can work directly with the coefficients rather than writing out full x-terms.

In synthetic division, you lay out the coefficients of the polynomial and bring down the leading coefficient as the first coefficient of the quotient. Then you repeatedly multiply the current quotient coefficient by c and add that product to the next coefficient. Repeat across all coefficients. The numbers you get in the process give you the quotient, and the final sum is the remainder. This single-pass procedure uses only the coefficients and the constant c, making it faster and with less writing than ordinary polynomial long division.

Other methods don’t fit this task as neatly. Polynomial long division can do the job, but it involves more algebraic clutter with x-terms and powers, so it’s slower. The Rational Zero Theorem helps identify possible roots, not perform division. The Quadratic Formula solves equations of degree two, not general division by a binomial like x - c.