What is the shortcut method for dividing a polynomial by a polynomial of the first degree?

What is the shortcut method for dividing a polynomial by a polynomial of the first degree?

• A. Synthetic Division
• B. Long Division
• C. Polynomial Long Division
• D. Division Algorithm

Answer: (A)

Synthetic division is the shortcut used when dividing by a linear polynomial (one of the form x minus a constant). The idea is that, instead of going through the full polynomial long division, you work directly with the coefficients, using a quick Horner-like process to get the quotient and remainder. How it works: take the coefficients of the dividend, bring down the leading coefficient as the first term of the quotient, then multiply that by the constant c from the divisor x minus c and add it to the next coefficient. Repeat the multiply-and-add step across all coefficients. The final number is the remainder, and the numbers you obtained along the way are the coefficients of the quotient. This is much faster than doing long division term by term. For example, dividing 2x^3 + 3x^2 − x + 5 by x − 4, you list the coefficients 2, 3, −1, 5, bring down the 2 as the first quotient coefficient, then follow the multiply-and-add steps with c = 4 to arrive at the quotient 2x^2 + 11x + 43 and remainder 177. Long division and polynomial long division are valid methods, but they are the more general, step-by-step approaches and don’t provide the same quick routine for linear divisors. The division algorithm states that a quotient and remainder exist for polynomial division, but it isn’t the specific shortcut used for linear divisors.

Synthetic division is the shortcut used when dividing by a linear polynomial (one of the form x minus a constant). The idea is that, instead of going through the full polynomial long division, you work directly with the coefficients, using a quick Horner-like process to get the quotient and remainder.

How it works: take the coefficients of the dividend, bring down the leading coefficient as the first term of the quotient, then multiply that by the constant c from the divisor x minus c and add it to the next coefficient. Repeat the multiply-and-add step across all coefficients. The final number is the remainder, and the numbers you obtained along the way are the coefficients of the quotient. This is much faster than doing long division term by term.

For example, dividing 2x^3 + 3x^2 − x + 5 by x − 4, you list the coefficients 2, 3, −1, 5, bring down the 2 as the first quotient coefficient, then follow the multiply-and-add steps with c = 4 to arrive at the quotient 2x^2 + 11x + 43 and remainder 177.

Long division and polynomial long division are valid methods, but they are the more general, step-by-step approaches and don’t provide the same quick routine for linear divisors. The division algorithm states that a quotient and remainder exist for polynomial division, but it isn’t the specific shortcut used for linear divisors.