Problem 41 - Beginning and Intermediate Algebra

Question

Explain when synthetic division may be used to divide polynomials.

Step-by-Step Solution

Answer

Synthetic division may be used to divide polynomials when the divisor is linear (of the form x-c, where c is a constant). It provides a more efficient approach compared to long division, working with the coefficients of the polynomial. For example, when dividing the polynomial \(P(x) = 2x^3 - 5x^2 + 3x + 1\) by the divisor \(x - 1\), the result would be \( (2x^2 - 3x + 1) + \dfrac{3}{x-1}\).

Step 1: Identify when synthetic division can be used

A synthetic division can be used to divide a polynomial provided that the divisor is linear or of the form x - c, where c is a constant term.

Step 2: Understand synthetic division method

In synthetic division, the coefficients of each term of the polynomial are written down in order. Then, the constant term c from the divisor x-c is used to perform the division.

Step 3: Performing synthetic division with an example

Let's divide the polynomial \( P(x) = 2x^3 - 5x^2 + 3x + 1 \) by the linear divisor \(x - 1\). 1. Write down the coefficients of the polynomial: 2, -5, 3, 1 2. Write the constant term from the divisor x - c: 1 3. Perform synthetic division as follows: ____________1______________ | 2 -5 3 1 +2 -3 -0 ----------- 0 1 3 4. Interpret the result obtained from synthetic division: The coefficients of the quotient are in the bottom row of our table: 2, -3, and 1. Since we started with a cubic polynomial and we are dividing by a linear polynomial, our result will be a quadratic polynomial. Thus, the quotient is: \( 2x^2 - 3x + 1 \) 5. Finally, the remainder of the division is 3, which can be written as a fraction over the divisor: \(\dfrac{3}{x-1}\) So, the result of the division is: \( (2x^2 - 3x + 1) + \dfrac{3}{x-1} \) This example demonstrated the usage of synthetic division for dividing a polynomial by a linear divisor. Synthetic division becomes handy when dealing with these specific types of divisions and is an efficient method compared to long division.