Problem 72 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Any problem that can be done by synthetic division can also be done by the method for long division of polynomials.

Step-by-Step Solution

Answer

The statement is true.

Step 1: Understanding Synthetic Division

Synthetic division is a shortcut method used in dividing a polynomial by a binomial of the form x - c. It decreases the amount of writing and calculations involved, especially when dealing with high power polynomials. However, it's limited in the sense that it does not work with divisors that are not of the form x-c.

Step 2: Understanding Long Division of Polynomials

Long division of polynomials is a more general method that can be used to divide any polynomial by another polynomial, regardless of the type of the divisor. It may involve more steps and calculations than synthetic division.

Step 3: Verifying the Statement

The statement says 'Any problem that can be done by synthetic division can also be done by the method for long division of polynomials.' Given that long division can handle more general cases than synthetic division, it is safe to say that problems solved by synthetic division (which are a subset) can also be solved by long division. So, the statement is true.