(x^3-8x^2+17x-10)/(x-5)

Jasonbradley

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·

Jun 17, 2024

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Dividing Polynomials: (x³ - 8x² + 17x - 10) / (x - 5)

This article will guide you through the process of dividing the polynomial (x³ - 8x² + 17x - 10) by (x - 5) using polynomial long division.

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to the long division of numbers. Here's how it works:

1. Set up the division: Write the dividend (x³ - 8x² + 17x - 10) inside the division symbol and the divisor (x - 5) outside.

        ___________
   x - 5 | x³ - 8x² + 17x - 10 
   

2. Divide the leading terms: Divide the leading term of the dividend (x³) by the leading term of the divisor (x). This gives us x². Write x² above the division symbol.

        x² _________
   x - 5 | x³ - 8x² + 17x - 10 
   

3. Multiply the divisor by the quotient term: Multiply (x - 5) by x². This gives us x³ - 5x². Write this below the dividend, aligning like terms.

        x² _________
   x - 5 | x³ - 8x² + 17x - 10 
           x³ - 5x² 
   

4. Subtract: Subtract the terms you just wrote from the dividend. This leaves us with -3x² + 17x.

        x² _________
   x - 5 | x³ - 8x² + 17x - 10 
           x³ - 5x²
           -------
               -3x² + 17x
   

5. Bring down the next term: Bring down the next term of the dividend (17x) to form the new polynomial.

        x² _________
   x - 5 | x³ - 8x² + 17x - 10 
           x³ - 5x²
           -------
               -3x² + 17x - 10
   

6. Repeat steps 2-5: Divide the leading term of the new polynomial (-3x²) by the leading term of the divisor (x). This gives us -3x. Write -3x above the division symbol.

        x² - 3x ______
   x - 5 | x³ - 8x² + 17x - 10 
           x³ - 5x²
           -------
               -3x² + 17x - 10
               -3x² + 15x
   

7. Continue repeating: Multiply (x - 5) by -3x. This gives us -3x² + 15x. Subtract this from the new polynomial, bring down the next term (-10), and repeat the process.

        x² - 3x ______
   x - 5 | x³ - 8x² + 17x - 10 
           x³ - 5x²
           -------
               -3x² + 17x - 10
               -3x² + 15x
               ---------
                       2x - 10
                       2x - 10 
   

8. Final step: Divide 2x by x, which gives us 2. Multiply (x - 5) by 2 and subtract from 2x - 10. We are left with a remainder of 0.

        x² - 3x + 2
   x - 5 | x³ - 8x² + 17x - 10 
           x³ - 5x²
           -------
               -3x² + 17x - 10
               -3x² + 15x
               ---------
                       2x - 10
                       2x - 10 
                       -------
                           0
   

Conclusion

Therefore, the result of dividing (x³ - 8x² + 17x - 10) by (x - 5) is x² - 3x + 2. There is no remainder. This means that (x - 5) is a factor of (x³ - 8x² + 17x - 10), and we can express the polynomial as:

(x³ - 8x² + 17x - 10) = (x - 5)(x² - 3x + 2)