(x4+4x3+16x−35)÷(x+5)

Jasonbradley

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Jun 17, 2024

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Dividing Polynomials: (x⁴ + 4x³ + 16x - 35) ÷ (x + 5)

This article will guide you through the process of dividing the polynomial (x⁴ + 4x³ + 16x - 35) by (x + 5) using polynomial long division.

Understanding Polynomial Long Division

Polynomial long division is a method for dividing polynomials, similar to the long division method used with integers. The process involves a series of steps to find the quotient and remainder of the division.

Step-by-Step Solution

1. Set up the division:

        ________
   x + 5 | x⁴ + 4x³ + 0x² + 16x - 35
   

2. 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 dividend.

        x³_______
   x + 5 | x⁴ + 4x³ + 0x² + 16x - 35 
   

3. Multiply the divisor by the quotient term:

   • Multiply (x + 5) by x³ to get x⁴ + 5x³.
   • Write the result below the dividend.

        x³_______
   x + 5 | x⁴ + 4x³ + 0x² + 16x - 35 
           x⁴ + 5x³
   

4. Subtract:

   • Subtract the product obtained in step 3 from the dividend.
   • Remember to change the signs of the terms being subtracted.

        x³_______
   x + 5 | x⁴ + 4x³ + 0x² + 16x - 35 
           x⁴ + 5x³
           -------
                -x³ + 0x² 
   

5. Bring down the next term:

   • Bring down the next term (0x²) from the dividend.

        x³_______
   x + 5 | x⁴ + 4x³ + 0x² + 16x - 35 
           x⁴ + 5x³
           -------
                -x³ + 0x² + 16x
   

6. Repeat steps 2-5:

   • Divide the leading term of the new dividend (-x³) by the leading term of the divisor (x). This gives us -x².
   • Write -x² above the x² term in the dividend.
   • Multiply (x + 5) by -x² to get -x³ - 5x².
   • Subtract the product from the dividend.
   • Bring down the next term (16x).

        x³ - x²____
   x + 5 | x⁴ + 4x³ + 0x² + 16x - 35 
           x⁴ + 5x³
           -------
                -x³ + 0x² + 16x
                -x³ - 5x²
                --------
                        5x² + 16x 
   

7. Repeat steps 2-5 again:

   • Divide the leading term of the new dividend (5x²) by the leading term of the divisor (x). This gives us 5x.
   • Write 5x above the x term in the dividend.
   • Multiply (x + 5) by 5x to get 5x² + 25x.
   • Subtract the product from the dividend.
   • Bring down the last term (-35).

        x³ - x² + 5x__
   x + 5 | x⁴ + 4x³ + 0x² + 16x - 35 
           x⁴ + 5x³
           -------
                -x³ + 0x² + 16x
                -x³ - 5x²
                --------
                        5x² + 16x - 35
                        5x² + 25x 
                        --------
                                -9x - 35
   

8. Final step:

   • Divide the leading term of the new dividend (-9x) by the leading term of the divisor (x). This gives us -9.
   • Write -9 above the constant term in the dividend.
   • Multiply (x + 5) by -9 to get -9x - 45.
   • Subtract the product from the dividend.

        x³ - x² + 5x - 9
   x + 5 | x⁴ + 4x³ + 0x² + 16x - 35 
           x⁴ + 5x³
           -------
                -x³ + 0x² + 16x
                -x³ - 5x²
                --------
                        5x² + 16x - 35
                        5x² + 25x 
                        --------
                                -9x - 35
                                -9x - 45
                                --------
                                        10
   

Therefore, the result of dividing (x⁴ + 4x³ + 16x - 35) by (x + 5) is:

(x⁴ + 4x³ + 16x - 35) ÷ (x + 5) = x³ - x² + 5x - 9 + 10/(x + 5)

The quotient is x³ - x² + 5x - 9 and the remainder is 10.