Using Long Division to Obtain the Polynomial 5x – 11 – 12x2 22x3 from x – 5

In algebra, polynomial division is a method used to divide one polynomial by another. This article will walk you through the process of obtaining the polynomial 5x – 11 – 12x2 22x3 through the polynomial division of (2x3 – 12x2 5x – 11) by (x – 5). This method is an essential skill for understanding polynomial functions and can be used in various applications ranging from engineering to linear algebra.

Understanding Polynomial Division

Polynomial division is a procedure that is essentially the same as long division of numbers. The process can be divided into several steps and is particularly useful when we need to simplify complex polynomials or factorize them.

Step 1: Setting Up the Division

The first step is to set up the division problem. In this case, we have the dividend (2x^3 - 12x^2 5x - 11) and the divisor (x - 5). We place them as follows:

Dividend: 2x3 – 12x2 5x – 11

Divisor: x - 5

Step 2: Perform the Division

Next, we perform the division step-by-step:

Divide the leading term of the dividend by the leading term of the divisor.

[ frac{2x^3}{x} 2x^2 ]

Write (2x^2) above the line.

Multiply the entire divisor by (2x^2) and subtract the result from the dividend.

[ (2x^3 - 12x^2 5x - 11) - (2x^3 - 1^2) -2x^2 5x - 11 ]

Repeat the process with the new polynomial (-2x^2 5x - 11).

Divide (-2x^2) by (x), which gives (-2x).

Multiply the entire divisor by (-2x) and subtract the result from (-2x^2 5x - 11).

[ (-2x^2 5x - 11) - (-2x^2 1) -5x - 11 ]

Repeat the process with the new polynomial (-5x - 11).

Divide (-5x) by (x), which gives (-5).

Multiply the entire divisor by (-5) and subtract the result from (-5x - 11).

[ (-5x - 11) - (-5x 25) -36 ]

Step 3: Write the Result

Putting it all together, we get:

[ frac{2x^3 - 12x^2 5x - 11}{x - 5} 2x^2 - 2x - 5 - frac{36}{x - 5} ]

Verification and Application

To verify, we can check the remainder and the coefficients of the quotient polynomial. From the division, we can express the polynomial as:

[ 2x^3 - 12x^2 5x - 11 (x - 5)(2x^2 - 2x - 5) - 36 ]

By expanding the right-hand side, we can ensure that it matches the original polynomial.

Conclusion

In this article, we’ve seen the step-by-step process of how to use polynomial division to obtain the polynomial 5x – 11 – 12x2 22x3 from x – 5. This method is essential for various algebraic operations and can be applied to simplify and solve complex polynomial equations.

Related Keywords

Polynomial division Long division PolyQuotient Theorem Polynomial remainder Polynomial multiplication

Further Reading

Understanding Polynomial Functions Applications of Polynomial Division

Conclusion

To summarize, mastering polynomial division allows you to manipulate polynomials more effectively. This skill is not only useful in algebra but also in various real-world applications. Throughout this article, we've walked through the process of obtaining the polynomial using polynomial division and verified the result.