%2F(x-3).jpeg "(6x^3+7x^2-x+26)/(x-3)")
Polynomial Division: (6x³ + 7x² - x + 26) / (x - 3)
This article will demonstrate the process of dividing the polynomial 6x³ + 7x² - x + 26 by (x - 3) using polynomial long division.
Understanding Polynomial Long Division
Polynomial long division is a method for dividing polynomials that mirrors the process of long division for numbers. Here's how it works:
- Set up the division: Write the dividend (6x³ + 7x² - x + 26) inside the division symbol and the divisor (x - 3) outside.
- Divide the leading terms: Divide the leading term of the dividend (6x³) by the leading term of the divisor (x). The result is 6x². Write this above the division symbol.
- Multiply and subtract: Multiply the quotient (6x²) by the divisor (x - 3). This gives 6x³ - 18x². Subtract this result from the dividend.
- Bring down the next term: Bring down the next term of the dividend (-x).
- Repeat steps 2-4: Now, divide the new leading term (13x²) by the leading term of the divisor (x). This gives 13x. Write this above the division symbol. Multiply 13x by (x - 3), subtract, and bring down the next term (26).
- Continue until the degree of the remainder is less than the degree of the divisor: Repeat steps 2-4 until the degree of the remainder is less than the degree of the divisor (in this case, a constant).
Step-by-Step Solution
Let's perform the division:
6x² + 13x + 38
x - 3 | 6x³ + 7x² - x + 26
-(6x³ - 18x²)
----------------
13x² - x
-(13x² - 39x)
-----------------
38x + 26
-(38x - 114)
-------------
140
Therefore, the result of the division is:
(6x³ + 7x² - x + 26) / (x - 3) = 6x² + 13x + 38 + 140/(x - 3)
Conclusion
Polynomial long division is a powerful tool for dividing polynomials. It allows us to break down complex expressions into simpler ones, which can be helpful in various mathematical applications.