Problem 67 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Every time I divide polynomials using synthetic division, I am using a highly condensed form of the long division procedure where omitting the variables and exponents does not involve the loss of any essential data.

Step-by-Step Solution

Answer

The statement is correct. Synthetic division is indeed a condensed form of long division and does not entail the loss of any essential data when omitting variables and exponents as they're inherently accounted for in the process.

Step 1: Verify the Statement

Firstly, confirm that synthetic division is indeed a condensed form of long division. This is factually true as synthetic division is just a shorthand method of dividing a polynomial by a linear binomial. It skips several steps involved in the long division process but yields the same result.

Step 2: Assess Exponent Omission

Next, evaluate whether omitting the exponents in synthetic division is without loss of any essential data. In synthetic division, although explicit exponents are not written in the process, their importance is implied as the coefficients are arranged in decreasing order of exponents. Hence, there is no loss of essential data. This means that the omitting of exponents in synthetic division does not result in any loss of essential information.

Step 3: Evaluate Variable Omission

Lastly, assess the impact of variable omission. In synthetic division, the variables are not expressly written but they are inherently part of the process as it involves coefficients of the variables in the dividend polynomial. Thus, despite not explicitly seeing the variables, synthesizing is based on those variables, hence, no crucial data is lost, reinforcing that the statement is correct. Hence, it can be concluded that the omission of variables and exponents in synthetic division does not result in the loss of any essential data.