%C3%B7(x%E2%88%926).jpeg "(x4−5x3−8x2+13x−12)÷(x−6)")
Dividing Polynomials: (x⁴ - 5x³ - 8x² + 13x - 12) ÷ (x - 6)
In this article, we'll explore the division of the polynomial (x⁴ - 5x³ - 8x² + 13x - 12) by (x - 6). We'll use polynomial long division to solve this problem.
Understanding Polynomial Long Division
Polynomial long division is similar to long division with numbers, but instead of digits, we are dealing with terms of polynomials. Here's a breakdown of the steps involved:
- Set up: Write the dividend (x⁴ - 5x³ - 8x² + 13x - 12) inside the division symbol and the divisor (x - 6) outside.
- Divide the leading terms: Divide the leading term of the dividend (x⁴) by the leading term of the divisor (x). The result (x³) becomes the first term of the quotient.
- Multiply: Multiply the divisor (x - 6) by the first term of the quotient (x³), resulting in (x⁴ - 6x³).
- Subtract: Subtract the product from the dividend.
- Bring down the next term: Bring down the next term (-8x²) from the dividend.
- Repeat: Repeat steps 2-5 with the new dividend (-x³ - 8x²). Continue this process until there are no more terms to bring down.
Performing the Division
Let's apply the steps to our problem:
x³ + x² - 2x + 1
______________________
x - 6 | x⁴ - 5x³ - 8x² + 13x - 12
-(x⁴ - 6x³)
-----------------
x³ - 8x²
-(x³ - 6x²)
--------------
-2x² + 13x
-(-2x² + 12x)
----------------
x - 12
-(x - 6)
------------
-6
Result
Therefore, the result of the division (x⁴ - 5x³ - 8x² + 13x - 12) ÷ (x - 6) is:
x³ + x² - 2x + 1 with a remainder of -6.
This can also be expressed as:
(x⁴ - 5x³ - 8x² + 13x - 12) ÷ (x - 6) = x³ + x² - 2x + 1 - 6/(x - 6)
Conclusion
By applying polynomial long division, we successfully divided (x⁴ - 5x³ - 8x² + 13x - 12) by (x - 6) and obtained the quotient x³ + x² - 2x + 1 with a remainder of -6. This process demonstrates a fundamental technique for working with polynomials and their division.