%2F(x%5E2%2B2x%2B3).jpeg "(2x^5+3x^4+8x^3+8x^2+18x+9)/(x^2+2x+3)")
Dividing Polynomials: (2x^5+3x^4+8x^3+8x^2+18x+9) / (x^2+2x+3)
This article will demonstrate the process of dividing the polynomial (2x^5+3x^4+8x^3+8x^2+18x+9) by (x^2+2x+3) using polynomial long division.
Polynomial Long Division
Polynomial long division is similar to regular long division, but instead of dividing numbers, we are dividing polynomials. Here's how it works:
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Set up the division problem: Write the dividend (2x^5+3x^4+8x^3+8x^2+18x+9) inside the division symbol and the divisor (x^2+2x+3) outside.
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Divide the leading terms: Divide the leading term of the dividend (2x^5) by the leading term of the divisor (x^2). This gives us 2x^3. Write this term above the division symbol.
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Multiply the divisor: Multiply the divisor (x^2+2x+3) by the term we just found (2x^3). This gives us 2x^5 + 4x^4 + 6x^3. Write this result below the dividend.
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Subtract: Subtract the result from step 3 from the dividend. This gives us -x^4 + 2x^3 + 8x^2 + 18x + 9.
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Bring down the next term: Bring down the next term of the dividend (18x).
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Repeat steps 2-5: Divide the new leading term (-x^4) by the leading term of the divisor (x^2). This gives us -x^2. Write this term above the division symbol.
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Continue the process: Repeat steps 2-5 until the degree of the remaining polynomial is less than the degree of the divisor.
Performing the Division
Here is the detailed division process:
2x^3 - x^2 + 4x + 2
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x^2+2x+3 | 2x^5 + 3x^4 + 8x^3 + 8x^2 + 18x + 9
-(2x^5 + 4x^4 + 6x^3)
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-x^4 + 2x^3 + 8x^2 + 18x
-(-x^4 - 2x^3 - 3x^2)
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4x^3 + 11x^2 + 18x
-(4x^3 + 8x^2 + 12x)
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3x^2 + 6x + 9
-(3x^2 + 6x + 9)
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0
Result
Therefore, the result of dividing (2x^5+3x^4+8x^3+8x^2+18x+9) by (x^2+2x+3) is 2x^3 - x^2 + 4x + 2.
Conclusion
Polynomial long division is a useful tool for dividing polynomials. By following the steps outlined above, you can successfully divide any polynomial by another polynomial of a lower or equal degree.