(x3+x2+x+2)÷(x2−1)

Jasonbradley

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Jun 17, 2024

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Performing Polynomial Long Division: (x³ + x² + x + 2) ÷ (x² - 1)

This article will guide you through the process of dividing the polynomial (x³ + x² + x + 2) by (x² - 1) using long division.

Understanding Polynomial Long Division

Polynomial long division is similar to the long division you learned in arithmetic. We are trying to find the quotient and remainder when dividing one polynomial by another. Here's how to approach it:

1. Set up the division: Write the dividend (x³ + x² + x + 2) inside the division symbol and the divisor (x² - 1) outside.

2. Focus on the leading terms: Divide the leading term of the dividend (x³) by the leading term of the divisor (x²). This gives us 'x'. Write 'x' above the x² term in the dividend.

3. Multiply the quotient by the divisor: Multiply the quotient (x) by the divisor (x² - 1) to get (x³ - x). Write this result below the dividend.

4. Subtract: Subtract the result from the dividend. This will leave you with (x² + x + 2).

5. Bring down the next term: Bring down the next term of the dividend (+2) to create the new polynomial (x² + x + 2).

6. Repeat steps 2-5: Now, divide the new leading term (x²) by the leading term of the divisor (x²), which gives us '1'. Write '1' next to the 'x' in the quotient. Multiply the quotient (x + 1) by the divisor (x² - 1), which gives us (x³ + x² - x - 1). Subtract this from the new dividend, leaving you with (2x + 3).

7. Stop when the degree of the remainder is less than the degree of the divisor: Since the degree of (2x + 3) is less than the degree of (x² - 1), we stop here.

Final Result

The result of the division is:

(x³ + x² + x + 2) ÷ (x² - 1) = x + 1 + (2x + 3)/(x² - 1)

This means that the quotient is (x + 1) and the remainder is (2x + 3).

Key points:

• Remainder: The remainder is always of a lower degree than the divisor.
• Quotient: The quotient represents the number of times the divisor goes into the dividend.
• Verification: You can check your answer by multiplying the quotient by the divisor and adding the remainder. This should give you the original dividend.