To solve the division of the polynomial [tex]\(6x^2 + x - 26\)[/tex] by [tex]\(x - 2\)[/tex] using synthetic division, follow these steps:
1. Identify the root of the divisor: Since we are dividing by [tex]\(x - 2\)[/tex], the root (or zero) of the divisor is [tex]\(2\)[/tex].
2. Set up the synthetic division: Write down the coefficients of the dividend polynomial [tex]\(6x^2 + x - 26\)[/tex], which are [tex]\(6\)[/tex], [tex]\(1\)[/tex], and [tex]\(-26\)[/tex]. Create a synthetic division table.
3. Perform the synthetic division:
- Start with the leading coefficient [tex]\(6\)[/tex] and bring it down.
- Multiply this coefficient by the root [tex]\(2\)[/tex] and write the result under the next coefficient.
- Add the values in this column and write the result below as the new coefficient.
- Repeat the multiplication and addition steps for each coefficient.
Here's the step-by-step process in table form:
[tex]\[
\begin{array}{r|rrr}
2 & 6 & 1 & -26 \\
& & 12 & 26 \\
\hline
& 6 & 13 & 0 \\
\end{array}
\][/tex]
- Step 1: Bring down the first coefficient [tex]\(6\)[/tex].
- Step 2: Multiply [tex]\(6\)[/tex] by [tex]\(2\)[/tex] (the root), which gives [tex]\(12\)[/tex]. Write this under the second coefficient [tex]\(1\)[/tex].
- Step 3: Add the second coefficient [tex]\(1\)[/tex] and [tex]\(12\)[/tex] to get [tex]\(13\)[/tex]. Write this under the line.
- Step 4: Multiply [tex]\(13\)[/tex] by [tex]\(2\)[/tex] to get [tex]\(26\)[/tex]. Write this under the third coefficient [tex]\(-26\)[/tex].
- Step 5: Add [tex]\(-26\)[/tex] and [tex]\(26\)[/tex] to get [tex]\(0\)[/tex]. Write this under the line.
4. Interpret the results:
- The numbers below the line [tex]\(6, 13, 0\)[/tex] represent the coefficients of the quotient polynomial and the remainder.
- The quotient polynomial is formed by the first two numbers, so the quotient is [tex]\(6x + 13\)[/tex].
- The remainder is the last number, which is [tex]\(0\)[/tex].
Thus, the result of dividing [tex]\((6x^2 + x - 26)\)[/tex] by [tex]\((x - 2)\)[/tex] is:
[tex]\[
6x + 13 \quad \text{with a remainder of} \quad 0.
\][/tex]
So,
[tex]\[
\left(6x^2 + x - 26\right) \div (x - 2) = 6x + 13.
\][/tex]
