Get the answers you've been searching for with IDNLearn.com. Our community is ready to provide in-depth answers and practical solutions to any questions you may have.
Sagot :
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].
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].
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your questions find answers at IDNLearn.com. Thanks for visiting, and come back for more accurate and reliable solutions.
