(x4+4x3+16x−35)÷(x+5)=

Jasonbradley

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Jun 17, 2024

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Polynomial Long Division: (x⁴+4x³+16x−35) ÷ (x+5)

This article will guide you through the process of performing polynomial long division to solve the problem: (x⁴+4x³+16x−35) ÷ (x+5)

Understanding Polynomial Long Division

Polynomial long division is a method for dividing polynomials, similar to long division with numbers. It helps us find the quotient and remainder when dividing one polynomial by another.

Steps for Polynomial Long Division

1. Set up the division:

   • Write the dividend (x⁴+4x³+16x−35) inside the division symbol.
   • Write the divisor (x+5) outside the division symbol.

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 division symbol, aligning it with the x⁴ term.

3. Multiply the quotient by the divisor:

   • Multiply x³ by the entire divisor (x+5). This gives us x⁴ + 5x³.
   • Write the result below the dividend, aligning like terms.

4. Subtract:

   • Subtract the result (x⁴ + 5x³) from the dividend.
   • This leaves us with -x³ + 16x - 35.

5. Bring down the next term:

   • Bring down the next term of the dividend (16x).

6. Repeat steps 2-5:

   • Divide the new leading term (-x³) by the leading term of the divisor (x). This gives us -x².
   • Write -x² above the division symbol, aligning it with the x³ term.
   • Multiply -x² by the divisor (x+5), giving us -x³ - 5x².
   • Subtract this result from the current expression.
   • Bring down the next term (-35).
   • Continue this process until the degree of the remainder is less than the degree of the divisor.

7. Write the answer:

   • The quotient is the polynomial above the division symbol.
   • The remainder is the last term in the division.

Applying the Steps

Let's apply these steps to our problem:

             x³ - x² + 5x - 9 
       ______________________
x+5 | x⁴ + 4x³ + 16x - 35 
      -(x⁴ + 5x³)
      -----------------
             -x³ + 16x
             -(-x³ - 5x²)
             -----------------
                    5x² + 16x
                    -(5x² + 25x)
                    -----------------
                           -9x - 35
                           -(-9x - 45)
                           -----------------
                                   10 

Conclusion

Therefore, (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. This result can be expressed as:

• Quotient + Remainder / Divisor

x³ - x² + 5x - 9 + 10/(x+5)