From personal advice to professional guidance, IDNLearn.com has the answers you seek. Get accurate and detailed answers to your questions from our dedicated community members who are always ready to help.
Sagot :
Let's solve the division of the polynomial [tex]\(\frac{x^4 - x^3 + x^2 - x + 2}{x - 2}\)[/tex] using polynomial long division.
1. Set up the division: Divide [tex]\(x^4 - x^3 + x^2 - x + 2\)[/tex] by [tex]\(x - 2\)[/tex].
[tex]\[ \begin{array}{r|rrrrr} x^3 + x^2 + 3x + 5 & x - 2 & \text{remainder}\\ \hline x^4 - x^3 + x^2 - x + 2 & & \end{array} \][/tex]
2. First term: Divide the leading term [tex]\(x^4\)[/tex] by the leading term [tex]\(x\)[/tex] of the divisor.
[tex]\[ x^4 \div x = x^3 \][/tex]
3. Multiply and subtract:
- Multiply [tex]\(x^3\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ x^3 \cdot (x - 2) = x^4 - 2x^3 \][/tex]
- Subtract this product from [tex]\(x^4 - x^3 + x^2 - x + 2\)[/tex]:
[tex]\[ (x^4 - x^3 + x^2 - x + 2) - (x^4 - 2x^3) = x^4 - x^3 + x^2 - x + 2 - x^4 + 2x^3 = x^3 + x^2 - x + 2 \][/tex]
4. Second term: Divide the new leading term [tex]\(x^3\)[/tex] by the leading term [tex]\(x\)[/tex] of the divisor.
[tex]\[ x^3 \div x = x^2 \][/tex]
5. Multiply and subtract:
- Multiply [tex]\(x^2\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ x^2 \cdot (x - 2) = x^3 - 2x^2 \][/tex]
- Subtract this product from [tex]\(x^3 + x^2 - x + 2\)[/tex]:
[tex]\[ (x^3 + x^2 - x + 2) - (x^3 - 2x^2) = x^3 + x^2 - x + 2 - x^3 + 2x^2 = 3x^2 - x + 2 \][/tex]
6. Third term: Divide the new leading term [tex]\(3x^2\)[/tex] by the leading term [tex]\(x\)[/tex] of the divisor.
[tex]\[ 3x^2 \div x = 3x \][/tex]
7. Multiply and subtract:
- Multiply [tex]\(3x\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ 3x \cdot (x - 2) = 3x^2 - 6x \][/tex]
- Subtract this product from [tex]\(3x^2 - x + 2\)[/tex]:
[tex]\[ (3x^2 - x + 2) - (3x^2 - 6x) = 3x^2 - x + 2 - 3x^2 + 6x = 5x + 2 \][/tex]
8. Fourth term: Divide the new leading term [tex]\(5x\)[/tex] by the leading term [tex]\(x\)[/tex] of the divisor.
[tex]\[ 5x \div x = 5 \][/tex]
9. Multiply and subtract:
- Multiply [tex]\(5\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ 5 \cdot (x - 2) = 5x - 10 \][/tex]
- Subtract this product from [tex]\(5x + 2\)[/tex]:
[tex]\[ (5x + 2) - (5x - 10) = 5x + 2 - 5x + 10 = 12 \][/tex]
10. Result: The quotient is [tex]\(x^3 + x^2 + 3x + 5\)[/tex] and the remainder is 12. Therefore, the division of [tex]\(\frac{x^4 - x^3 + x^2 - x + 2}{x - 2}\)[/tex] results in:
[tex]\[ \frac{x^4 - x^3 + x^2 - x + 2}{x - 2} = x^3 + x^2 + 3x + 5 + \frac{12}{x - 2} \][/tex]
1. Set up the division: Divide [tex]\(x^4 - x^3 + x^2 - x + 2\)[/tex] by [tex]\(x - 2\)[/tex].
[tex]\[ \begin{array}{r|rrrrr} x^3 + x^2 + 3x + 5 & x - 2 & \text{remainder}\\ \hline x^4 - x^3 + x^2 - x + 2 & & \end{array} \][/tex]
2. First term: Divide the leading term [tex]\(x^4\)[/tex] by the leading term [tex]\(x\)[/tex] of the divisor.
[tex]\[ x^4 \div x = x^3 \][/tex]
3. Multiply and subtract:
- Multiply [tex]\(x^3\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ x^3 \cdot (x - 2) = x^4 - 2x^3 \][/tex]
- Subtract this product from [tex]\(x^4 - x^3 + x^2 - x + 2\)[/tex]:
[tex]\[ (x^4 - x^3 + x^2 - x + 2) - (x^4 - 2x^3) = x^4 - x^3 + x^2 - x + 2 - x^4 + 2x^3 = x^3 + x^2 - x + 2 \][/tex]
4. Second term: Divide the new leading term [tex]\(x^3\)[/tex] by the leading term [tex]\(x\)[/tex] of the divisor.
[tex]\[ x^3 \div x = x^2 \][/tex]
5. Multiply and subtract:
- Multiply [tex]\(x^2\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ x^2 \cdot (x - 2) = x^3 - 2x^2 \][/tex]
- Subtract this product from [tex]\(x^3 + x^2 - x + 2\)[/tex]:
[tex]\[ (x^3 + x^2 - x + 2) - (x^3 - 2x^2) = x^3 + x^2 - x + 2 - x^3 + 2x^2 = 3x^2 - x + 2 \][/tex]
6. Third term: Divide the new leading term [tex]\(3x^2\)[/tex] by the leading term [tex]\(x\)[/tex] of the divisor.
[tex]\[ 3x^2 \div x = 3x \][/tex]
7. Multiply and subtract:
- Multiply [tex]\(3x\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ 3x \cdot (x - 2) = 3x^2 - 6x \][/tex]
- Subtract this product from [tex]\(3x^2 - x + 2\)[/tex]:
[tex]\[ (3x^2 - x + 2) - (3x^2 - 6x) = 3x^2 - x + 2 - 3x^2 + 6x = 5x + 2 \][/tex]
8. Fourth term: Divide the new leading term [tex]\(5x\)[/tex] by the leading term [tex]\(x\)[/tex] of the divisor.
[tex]\[ 5x \div x = 5 \][/tex]
9. Multiply and subtract:
- Multiply [tex]\(5\)[/tex] by [tex]\(x - 2\)[/tex]:
[tex]\[ 5 \cdot (x - 2) = 5x - 10 \][/tex]
- Subtract this product from [tex]\(5x + 2\)[/tex]:
[tex]\[ (5x + 2) - (5x - 10) = 5x + 2 - 5x + 10 = 12 \][/tex]
10. Result: The quotient is [tex]\(x^3 + x^2 + 3x + 5\)[/tex] and the remainder is 12. Therefore, the division of [tex]\(\frac{x^4 - x^3 + x^2 - x + 2}{x - 2}\)[/tex] results in:
[tex]\[ \frac{x^4 - x^3 + x^2 - x + 2}{x - 2} = x^3 + x^2 + 3x + 5 + \frac{12}{x - 2} \][/tex]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your search for solutions ends here at IDNLearn.com. Thank you for visiting, and come back soon for more helpful information.