%2F(x%5E2%2B1).jpeg "(7x^3+11x^2+7x+5)/(x^2+1)")
Performing Polynomial Long Division: (7x^3+11x^2+7x+5)/(x^2+1)
This article will guide you through the process of dividing the polynomial 7x³ + 11x² + 7x + 5 by x² + 1 using polynomial long division.
Understanding Polynomial Long Division
Polynomial long division is similar to the long division you learned in elementary school, but instead of dividing numbers, we're dividing polynomials. The goal is to find the quotient and remainder of the division.
Steps for Performing the Division
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Set up the division:
- Write the dividend (7x³ + 11x² + 7x + 5) inside the division symbol.
- Write the divisor (x² + 1) outside the division symbol.
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Focus on the leading terms:
- Divide the leading term of the dividend (7x³) by the leading term of the divisor (x²). This gives you 7x.
- Write 7x above the division symbol, aligning it with the x³ term.
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Multiply the divisor by the quotient term:
- Multiply (x² + 1) by 7x, which gives you 7x³ + 7x.
- Write this result below the dividend, aligning the terms.
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Subtract:
- Subtract the result from the dividend. This will cancel out the 7x³ term and leave you with 11x² + 0x + 5.
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Bring down the next term:
- Bring down the next term of the dividend (5) to the bottom of the expression.
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Repeat steps 2-5:
- Now focus on the new leading term (11x²).
- Divide 11x² by x² to get 11.
- Write 11 next to 7x above the division symbol.
- Multiply (x² + 1) by 11, which gives you 11x² + 11.
- Subtract this result from the current expression, canceling out the 11x² term.
- This leaves you with -6.
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The remainder:
- Since the degree of the remainder (-6) is less than the degree of the divisor (x² + 1), we stop the division.
Result
Therefore, the result of dividing (7x³ + 11x² + 7x + 5) by (x² + 1) is:
Quotient: 7x + 11 Remainder: -6
This can be written as:
(7x³ + 11x² + 7x + 5) / (x² + 1) = 7x + 11 - 6/(x² + 1)
This means that the original polynomial can be expressed as the product of the divisor and the quotient plus the remainder.