%2F(x%2B7).jpeg "(4x^3+27x^2+3x+64)/(x+7)")
Polynomial Long Division: (4x^3 + 27x^2 + 3x + 64) / (x + 7)
This article will guide you through the process of dividing the polynomial 4x³ + 27x² + 3x + 64 by x + 7 using long division.
1. Setting Up the Division
First, set up the long division problem like you would with regular numbers:
__________
x + 7 | 4x³ + 27x² + 3x + 64
2. Dividing the Leading Terms
- Focus on the leading terms: Divide the leading term of the dividend (4x³) by the leading term of the divisor (x).
- Write the result above the line: 4x³ / x = 4x², so write 4x² above the line.
4x²
x + 7 | 4x³ + 27x² + 3x + 64
3. Multiply and Subtract
- Multiply the divisor by the term you just wrote: (x + 7) * 4x² = 4x³ + 28x²
- Write the result below the dividend:
- Subtract the result from the dividend:
4x²
x + 7 | 4x³ + 27x² + 3x + 64
-(4x³ + 28x²)
-x² + 3x
4. Repeat the Process
- Bring down the next term: Bring down the +3x term.
- Repeat steps 2 and 3: Divide the new leading term (-x²) by the leading term of the divisor (x).
- Write the result above the line: -x² / x = -x, so write -x above the line.
- Multiply the divisor by -x: (x + 7) * -x = -x² - 7x.
- Subtract the result:
4x² - x
x + 7 | 4x³ + 27x² + 3x + 64
-(4x³ + 28x²)
-x² + 3x
-(-x² - 7x)
10x
5. Bring Down and Repeat
- Bring down the last term: Bring down the +64.
- Repeat steps 2 and 3: Divide the new leading term (10x) by the leading term of the divisor (x).
- Write the result above the line: 10x / x = 10, so write 10 above the line.
- Multiply the divisor by 10: (x + 7) * 10 = 10x + 70.
- Subtract the result:
4x² - x + 10
x + 7 | 4x³ + 27x² + 3x + 64
-(4x³ + 28x²)
-x² + 3x
-(-x² - 7x)
10x + 64
-(10x + 70)
-6
6. The Result
The final result of the division is:
4x² - x + 10 - 6/(x + 7)
Therefore, (4x³ + 27x² + 3x + 64) / (x + 7) = 4x² - x + 10 - 6/(x + 7)