Problem 24 - College Algebra

Question

Divide using synthetic division. $$\left(x^{5}+4 x^{4}-3 x^{2}+2 x+3\right) \div(x-3)$$

Step-by-Step Solution

Answer

The result of the synthetic division is \(x^{4}+7x^{3}+21x^{2}+66x+198+\frac{600}{x-3}\)

Step 1: Setup the Synthetic Division

In this step, set up the polynomial for synthetic division. Write the coefficients of the polynomial in descending order of the powers of \(x\) (replacing missing powers with a zero). Also, write the number we're dividing by \( (x-3)\), which will be the number \(3\). So, we write the numbers as follows: \(3 | 1,4,0,-3,2,3 \)

Step 2: Perform Synthetic Division

The first coefficient \(1\) is brought down as the first coefficient of the quotient. Multiply the new number in the bottom row by the number we're dividing by (3 in this case), and write this product under the second number in the top row. Sum up these numbers. Repeat this process of multiplying and adding for all coefficients in the top row. We get the following numbers: \(3 | 1, 7, 21, 66, 198, 600\)

Step 3: Write the Quotient Polynomial

Notice that we have lost a degree in our polynomial, it's a degree 4 polynomial now. The proper notation for the resulting polynomial is \(x^{4}+7x^{3}+21x^{2}+66x+198\), and the remaining number \(600\) is our remainder. Therefore the result is the polynomial plus the remainder divided by \(x-3\).