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Let's perform the division of the polynomial [tex]\( 4x^5 - 2x^3 + 5x^2 - 6x + 1 \)[/tex] by [tex]\( x - 2 \)[/tex] using synthetic division.
1. Identify the coefficients of the polynomial: [tex]\( 4, 0, -2, 5, -6, 1 \)[/tex].
- Note that there is no [tex]\( x^4 \)[/tex] term, so its coefficient is 0.
2. The divisor is [tex]\( x - 2 \)[/tex], so the number to use in synthetic division is 2.
3. Write down the coefficients:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \end{array} \][/tex]
4. Start the synthetic division:
- Bring down the first coefficient (4) as it is:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 \\ \end{array} \][/tex]
- Multiply 2 by the result (4), which gives 8. Write this below the second coefficient:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 \\ \end{array} \][/tex]
- Add the second coefficient (0) and 8:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 \\ & & 8 \\ \end{array} \][/tex]
- Multiply 2 by 8, which gives 16. Write this below the third coefficient:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 & 16 \\ & & 8 & \\ \end{array} \][/tex]
- Add the third coefficient (-2) and 16:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 & 16 \\ & & 8 & 14 \\ \end{array} \][/tex]
- Multiply 2 by 14, which gives 28. Write this below the fourth coefficient:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 & 16 & 28 \\ & & 8 & 14 & \\ \end{array} \][/tex]
- Add the fourth coefficient (5) and 28:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 & 16 & 28 \\ & & 8 & 14 & 33 \\ \end{array} \][/tex]
- Multiply 2 by 33, which gives 66. Write this below the fifth coefficient:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 & 16 & 28 & 66 \\ & & 8 & 14 & 33 & \\ \end{array} \][/tex]
- Add the fifth coefficient (-6) and 66:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 & 16 & 28 & 66 \\ & & 8 & 14 & 33 & 60 \\ \end{array} \][/tex]
- Multiply 2 by 60, which gives 120. Write this below the sixth coefficient:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 & 16 & 28 & 66 & 120 \\ & & 8 & 14 & 33 & 60 \\ \end{array} \][/tex]
- Add the sixth coefficient (1) and 120:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 & 16 & 28 & 66 & 120 \\ & & 8 & 14 & 33 & 60 & 121 \\ \end{array} \][/tex]
5. The result of the synthetic division gives the coefficients of the quotient and the remainder. The quotient is [tex]\( 4x^4 + 8x^3 + 14x^2 + 33x + 60 \)[/tex] and the remainder is 121.
So, the division of [tex]\( 4x^5 - 2x^3 + 5x^2 - 6x + 1 \)[/tex] by [tex]\( x - 2 \)[/tex] results in:
[tex]\[ \frac{4x^5 - 2x^3 + 5x^2 - 6x + 1}{x-2} = 4x^4 + 8x^3 + 14x^2 + 33x + 60 + \frac{121}{x-2} \][/tex]
1. Identify the coefficients of the polynomial: [tex]\( 4, 0, -2, 5, -6, 1 \)[/tex].
- Note that there is no [tex]\( x^4 \)[/tex] term, so its coefficient is 0.
2. The divisor is [tex]\( x - 2 \)[/tex], so the number to use in synthetic division is 2.
3. Write down the coefficients:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \end{array} \][/tex]
4. Start the synthetic division:
- Bring down the first coefficient (4) as it is:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 \\ \end{array} \][/tex]
- Multiply 2 by the result (4), which gives 8. Write this below the second coefficient:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 \\ \end{array} \][/tex]
- Add the second coefficient (0) and 8:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 \\ & & 8 \\ \end{array} \][/tex]
- Multiply 2 by 8, which gives 16. Write this below the third coefficient:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 & 16 \\ & & 8 & \\ \end{array} \][/tex]
- Add the third coefficient (-2) and 16:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 & 16 \\ & & 8 & 14 \\ \end{array} \][/tex]
- Multiply 2 by 14, which gives 28. Write this below the fourth coefficient:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 & 16 & 28 \\ & & 8 & 14 & \\ \end{array} \][/tex]
- Add the fourth coefficient (5) and 28:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 & 16 & 28 \\ & & 8 & 14 & 33 \\ \end{array} \][/tex]
- Multiply 2 by 33, which gives 66. Write this below the fifth coefficient:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 & 16 & 28 & 66 \\ & & 8 & 14 & 33 & \\ \end{array} \][/tex]
- Add the fifth coefficient (-6) and 66:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 & 16 & 28 & 66 \\ & & 8 & 14 & 33 & 60 \\ \end{array} \][/tex]
- Multiply 2 by 60, which gives 120. Write this below the sixth coefficient:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 & 16 & 28 & 66 & 120 \\ & & 8 & 14 & 33 & 60 \\ \end{array} \][/tex]
- Add the sixth coefficient (1) and 120:
[tex]\[ \begin{array}{c|ccccc} 2 & 4 & 0 & -2 & 5 & -6 & 1 \\ \hline & 4 & 8 & 16 & 28 & 66 & 120 \\ & & 8 & 14 & 33 & 60 & 121 \\ \end{array} \][/tex]
5. The result of the synthetic division gives the coefficients of the quotient and the remainder. The quotient is [tex]\( 4x^4 + 8x^3 + 14x^2 + 33x + 60 \)[/tex] and the remainder is 121.
So, the division of [tex]\( 4x^5 - 2x^3 + 5x^2 - 6x + 1 \)[/tex] by [tex]\( x - 2 \)[/tex] results in:
[tex]\[ \frac{4x^5 - 2x^3 + 5x^2 - 6x + 1}{x-2} = 4x^4 + 8x^3 + 14x^2 + 33x + 60 + \frac{121}{x-2} \][/tex]
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