Questions: Use the long division method to find the result when 2x^3 - 6x^2 + x + 9 is divided by x + 1. If there is a remainder, express the result in the form q(x) + r(x)/b(x).

Use the long division method to find the result when 2x^3 - 6x^2 + x + 9 is divided by x + 1. If there is a remainder, express the result in the form q(x) + r(x)/b(x).

Transcript text: Use the long division method to find the result when $2 x^{3}-6 x^{2}+x+9$ is divided by $x+1$. If there is a remainder, express the result in the form $q(x)+\frac{r(x)}{b(x)}$.

Solution

Solution Steps

Step 1: Setup

Given the dividend polynomial $P(x)$ and the divisor polynomial $D(x)$, we write them in standard form ensuring all terms are present. $P(x) = 2x^3 - 6x^2 + x^1 + 9x^0$ $D(x) = x^1 + x^0$

Step 2: Division Step

The leading term of $P(x)$ is divided by the leading term of $D(x)$ to find the first term of the quotient $q(x)$.

Step 3: Multiply and Subtract

Multiply $D(x)$ by the term just found and subtract the result from $P(x)$ to get a new polynomial.

Step 4: Repeat

This process is repeated, treating the new polynomial as the dividend, until the degree of the remainder is less than the degree of $D(x)$.