To find the remainder when dividing the polynomial [tex]\(3x^5 - 4x^3 + x^2 + 1\)[/tex] by the factor [tex]\((x + 2)\)[/tex] using synthetic division, follow these steps:
Step 1: Set up synthetic division.
- Write down the coefficients of the polynomial. Since there are missing terms, include coefficients of 0 for these terms.
The polynomial [tex]\(3x^5 - 4x^3 + x^2 + 1\)[/tex] has coefficients [3, 0, -4, 1, 0, 1].
- The factor [tex]\(x + 2\)[/tex] means we use [tex]\(c = -2\)[/tex] for synthetic division.
Step 2: Perform synthetic division.
1. Write down the coefficients: [tex]\(3, 0, -4, 1, 0, 1\)[/tex].
2. Bring down the first coefficient (3) as it is.
3. Multiply this number by [tex]\(c = -2\)[/tex] and write the result under the next coefficient.
4. Add the number from the previous step to the next coefficient and write the sum below the line.
5. Repeat the multiplication and addition steps until you fill the row.
Here is the step-by-step division:
[tex]\[
\begin{array}{r|rrrrrr}
-2 & 3 & 0 & -4 & 1 & 0 & 1 \\
& & -6 & 12 & -16 & 30 & -60 \\
\hline
& 3 & -6 & 8 & -15 & 30 & -59 \\
\end{array}
\][/tex]
Step 3: Interpret the results.
- The numbers at the bottom (3, -6, 8, -15, 30) are the coefficients of the quotient polynomial.
- The last number, [tex]\(-59\)[/tex], is the remainder.
So, the remainder when [tex]\(3x^5 - 4x^3 + x^2 + 1\)[/tex] is divided by [tex]\((x + 2)\)[/tex] is [tex]\(-59\)[/tex].
Hence, the correct answer is [tex]\(-59\)[/tex].
