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(10x^3+5x^2-1) Divided By (2x^3-4x^2-x+2)

4 min read Jun 16, 2024
(10x^3+5x^2-1) Divided By (2x^3-4x^2-x+2)

Dividing Polynomials: (10x³ + 5x² - 1) ÷ (2x³ - 4x² - x + 2)

This article will guide you through the process of dividing the polynomial 10x³ + 5x² - 1 by 2x³ - 4x² - 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 instead of dealing with numbers, we're working with polynomials. The goal is to find the quotient and remainder of dividing one polynomial by another.

Step-by-Step Process

  1. Set up the division:

        ____________
2x³-4x²-x+2 | 10x³ + 5x² - 1

  2. Divide the leading terms:

    • The leading term of the divisor (2x³) goes into the leading term of the dividend (10x³) 5 times.
    • Write 5 above the dividend, and multiply the divisor (2x³ - 4x² - x + 2) by 5.
    • Write the result (10x³ - 20x² - 5x + 10) below the dividend.
        5          
    ____________
2x³-4x²-x+2 | 10x³ + 5x² - 1 
                10x³ - 20x² - 5x + 10

  3. Subtract:

    • Subtract the expression we just wrote from the dividend.
    • Remember to change the signs of the terms being subtracted.
        5          
    ____________
2x³-4x²-x+2 | 10x³ + 5x² - 1 
                10x³ - 20x² - 5x + 10 
                --------------------
                        25x² + 5x - 11

  4. Bring down the next term:

    • Bring down the next term (-1) from the dividend.
        5          
    ____________
2x³-4x²-x+2 | 10x³ + 5x² - 1 
                10x³ - 20x² - 5x + 10 
                --------------------
                        25x² + 5x - 11

  5. Repeat steps 2-4:

    • Now, focus on the new leading term (25x²) and the leading term of the divisor (2x³).
    • 25x² goes into 2x³ (25/2)x times.
    • Write (25/2)x above the dividend, multiply the divisor by (25/2)x, and subtract.
        5 + (25/2)x      
    ____________
2x³-4x²-x+2 | 10x³ + 5x² - 1 
                10x³ - 20x² - 5x + 10 
                --------------------
                        25x² + 5x - 11 
                        25x² - 50x - (25/2)x + 25 
                        -------------------------
                                  55x + (25/2)x - 36

  6. Continue the process:

    • Repeat steps 2-4 until the degree of the remaining polynomial is less than the degree of the divisor.
    • You might need to continue a few more iterations, depending on the specific polynomials.

  7. Final result:

    • Once the degree of the remaining polynomial is less than the degree of the divisor, this remaining polynomial is the remainder.
    • The polynomial above the division line is the quotient.

Conclusion

Therefore, the result of dividing (10x³ + 5x² - 1) by (2x³ - 4x² - x + 2) is:

Quotient: 5 + (25/2)x

Remainder: 55x + (25/2)x - 36

This can be expressed as:

(10x³ + 5x² - 1) = (2x³ - 4x² - x + 2)(5 + (25/2)x) + (55x + (25/2)x - 36)

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