To divide [tex]\( 7x^3 - 64x^2 + 67x - 24 \)[/tex] by [tex]\( x - 8 \)[/tex] using synthetic division, follow these steps:
1. Set up the synthetic division:
- Write down the coefficients of the polynomial [tex]\( 7x^3 - 64x^2 + 67x - 24 \)[/tex]. These are 7, -64, 67, and -24.
- The divisor [tex]\( x - 8 \)[/tex] means we use the value [tex]\( 8 \)[/tex] for the synthetic division process.
2. Perform the synthetic division:
- Write down the divisor value [tex]\( 8 \)[/tex] on the left side.
- Write the coefficients [tex]\( 7, -64, 67, -24 \)[/tex] in a row.
[tex]\[
\begin{array}{r|rrrr}
8 & 7 & -64 & 67 & -24 \\
\hline
& 7 & & & \\
\end{array}
\][/tex]
3. Bring down the first coefficient (7):
- Place [tex]\( 7 \)[/tex] directly under the line.
[tex]\[
\begin{array}{r|rrrr}
8 & 7 & -64 & 67 & -24 \\
\hline
& 7 & & & \\
\end{array}
\][/tex]
4. Multiply and add:
- Multiply the divisor [tex]\( 8 \)[/tex] by the number just written below the line (7). Place the result below the next coefficient.
[tex]\[
\begin{array}{r|rrrr}
8 & 7 & -64 & 67 & -24 \\
\hline
& 7 & 56 & & \\
\end{array}
\][/tex]
- Add [tex]\(-64\)[/tex] and [tex]\( 56 \)[/tex] to get [tex]\( -8 \)[/tex]. Write this result under the line.
[tex]\[
\begin{array}{r|rrrr}
8 & 7 & -64 & 67 & -24 \\
\hline
& 7 & -8 & & \\
\end{array}
\][/tex]
- Repeat this process: Multiply [tex]\( -8 \)[/tex] by [tex]\( 8 \)[/tex] and place the result under the next coefficient.
[tex]\[
\begin{array}{r|rrrr}
8 & 7 & -64 & 67 & -24 \\
\hline
& 7 & -8 & -64 & \\
\end{array}
\][/tex]
- Add [tex]\( 67 \)[/tex] and [tex]\( -64 \)[/tex] to get [tex]\( 3 \)[/tex]. Write this result under the line.
[tex]\[
\begin{array}{r|rrrr}
8 & 7 & -64 & 67 & -24 \\
\hline
& 7 & -8 & 3 & \\
\end{array}
\][/tex]
- Repeat once more: Multiply [tex]\( 3 \)[/tex] by [tex]\( 8 \)[/tex] and place the result under the last coefficient.
[tex]\[
\begin{array}{r|rrrr}
8 & 7 & -64 & 67 & -24 \\
\hline
& 7 & -8 & 3 & 24 \\
\end{array}
\][/tex]
- Add [tex]\( -24 \)[/tex] and [tex]\( 24 \)[/tex] to get [tex]\( 0 \)[/tex]. Write this result under the line.
[tex]\[
\begin{array}{r|rrrr}
8 & 7 & -64 & 67 & -24 \\
\hline
& 7 & -8 & 3 & 0 \\
\end{array}
\][/tex]
5. Interpret the results:
- The numbers written below the line, except the last one, are the coefficients of the quotient polynomial. The last number is the remainder.
- Therefore, the quotient is [tex]\( 7x^2 - 8x + 3 \)[/tex] and the remainder is [tex]\( 0 \)[/tex].
6. Conclusion:
The synthetic division of [tex]\( 7x^3 - 64x^2 + 67x - 24 \)[/tex] by [tex]\( x - 8 \)[/tex] gives:
[tex]\[ 7x^3 - 64x^2 + 67x - 24 = (x - 8)(7x^2 - 8x + 3) + 0 \][/tex]
So, the quotient is [tex]\( 7x^2 - 8x + 3 \)[/tex] and the remainder is [tex]\( 0 \)[/tex].
