-%C3%B7-(x-%E2%80%93-2).jpeg "(2x4 – 3x3 – 6x2 + 11x + 8) ÷ (x – 2)")
Dividing Polynomials: (2x⁴ – 3x³ – 6x² + 11x + 8) ÷ (x – 2)
This article will guide you through the process of dividing the polynomial (2x⁴ – 3x³ – 6x² + 11x + 8) by the binomial (x – 2) using polynomial long division.
Understanding Polynomial Long Division
Polynomial long division is similar to the long division you learned in elementary school, but with polynomials instead of numbers. The goal is to find the quotient and remainder when one polynomial is divided by another.
Steps for Polynomial Long Division
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Set up the division: Write the dividend (2x⁴ – 3x³ – 6x² + 11x + 8) inside the division symbol and the divisor (x – 2) outside.
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Divide the leading terms: Divide the leading term of the dividend (2x⁴) by the leading term of the divisor (x). This gives us 2x³. Write 2x³ above the division symbol.
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Multiply and subtract: Multiply the divisor (x – 2) by 2x³ and write the result (2x⁴ – 4x³) below the dividend. Subtract this result from the dividend.
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Bring down the next term: Bring down the next term (-6x²) from the dividend.
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Repeat steps 2-4: Repeat the process of dividing the leading term, multiplying, and subtracting. In this case, we divide -x³ by x, getting -x². We multiply (x – 2) by -x² and subtract.
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Continue until the degree of the remainder is less than the degree of the divisor: Keep repeating the process until the degree of the remainder is less than the degree of the divisor.
Performing the Division
Following the steps above:
2x³ - x² - 8x - 5
x - 2 | 2x⁴ - 3x³ - 6x² + 11x + 8
2x⁴ - 4x³
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-x³ - 6x²
-x³ + 2x²
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-8x² + 11x
-8x² + 16x
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-5x + 8
-5x + 10
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-2
Conclusion
Therefore, (2x⁴ – 3x³ – 6x² + 11x + 8) ÷ (x – 2) equals 2x³ – x² – 8x – 5 with a remainder of -2. This can be expressed as:
(2x⁴ – 3x³ – 6x² + 11x + 8) = (x – 2)(2x³ – x² – 8x – 5) - 2
This process demonstrates how to divide polynomials using long division, a fundamental skill in algebra and calculus.