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Dividing Polynomials: (x⁴ - x² - 7) ÷ (x + 4)
This article will guide you through the process of dividing the polynomial (x⁴ - x² - 7) by (x + 4) using polynomial long division.
Understanding Polynomial Long Division
Polynomial long division is similar to long division with numbers. We aim to find the quotient and remainder when dividing one polynomial by another.
Steps for Polynomial Long Division
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Set up the division: Write the dividend (x⁴ - x² - 7) inside the division symbol and the divisor (x + 4) outside. Note that we need to include placeholders for missing terms (like x³ and x¹) in the dividend, so we rewrite it as x⁴ + 0x³ - x² + 0x - 7.
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Divide the leading terms: Divide the leading term of the dividend (x⁴) by the leading term of the divisor (x). This gives us x³. Write x³ above the x³ term in the quotient.
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Multiply the quotient by the divisor: Multiply x³ by (x + 4) to get x⁴ + 4x³.
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Subtract: Subtract (x⁴ + 4x³) from the dividend. This gives us -4x³ - x².
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Bring down the next term: Bring down the next term (-x²) from the dividend.
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Repeat steps 2-5: Now, divide the new leading term (-4x³) by the leading term of the divisor (x). This gives us -4x². Write -4x² above the x² term in the quotient.
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Multiply: Multiply -4x² by (x + 4) to get -4x³ - 16x².
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Subtract: Subtract (-4x³ - 16x²) from the current expression. This gives us 15x² + 0x.
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Repeat: Bring down the next term (0x). Divide the new leading term (15x²) by the leading term of the divisor (x). This gives us 15x. Write 15x above the x term in the quotient.
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Multiply: Multiply 15x by (x + 4) to get 15x² + 60x.
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Subtract: Subtract (15x² + 60x) from the current expression. This gives us -60x - 7.
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Repeat: Bring down the next term (-7). Divide the new leading term (-60x) by the leading term of the divisor (x). This gives us -60. Write -60 above the constant term in the quotient.
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Multiply: Multiply -60 by (x + 4) to get -60x - 240.
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Subtract: Subtract (-60x - 240) from the current expression. This gives us 233.
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The remainder: The remainder is 233.
The Final Result
Therefore, the result of dividing (x⁴ - x² - 7) by (x + 4) is:
(x⁴ - x² - 7) ÷ (x + 4) = x³ - 4x² + 15x - 60 + 233/(x + 4)
Conclusion
Polynomial long division can be used to divide polynomials of any degree. This process involves systematically dividing, multiplying, and subtracting terms until a remainder is obtained. Remember to include placeholders for missing terms in the dividend to ensure accurate division.