Problem 31 - College Algebra

Question

Divide using synthetic division. $$\frac{2 x^{5}-3 x^{4}+x^{3}-x^{2}+2 x-1}{x+2}$$

Step-by-Step Solution

Answer

The quotient of the division \( \frac{2 x^{5}-3 x^{4}+x^{3}-x^{2}+2 x-1}{x+2} \) is \( 2x^{4}+x^{3}-x^{2}+3x+7 -\frac{15}{x+2} \)

Step 1: Setup

In synthetic division, the coefficients of the polynomial (including zero for missing terms) are written in a row. In this case, the given polynomial is \(2 x^{5}-3 x^{4}+x^{3}-x^{2}+2 x-1\), hence the coefficients are 2, -3, 1, -1, 2, -1. The right side of synthethic division scheme must be the coefficient of the \(x\) term of the divisor inverted, in this case \(x+ 2\), hence -2.

Step 2: Synthetic Division Process Part I

The first thing you have to do is drop the leading coefficient, which is 2, straight down. Multiply this by -2 (the value in the rightmost column), and write the result underneath the next coefficient in the second row. This operation will be performed for the rest of the coefficients.

Step 3: Synthetic Division Process Part II

Next, add vertically in each column. The new result becomes the coefficient for the corresponding term in the quotient polynomial. Repeat the process: Multiply, place it in the next column underneath, then add, until there are no further columns.

Step 4: Write the Result

Finally, you write down the resulting coefficients as the terms of the quotient polynomial. The degree of the quotient polynomial will always be one less than the degree of the original polynomial, as long as you divided by something of degree 1.