Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use long division to divide. $$\frac{2 x^{3}-4 x^{2}-15 x+5}{(x-1)^{2}}$$

Step-by-Step Solution

Answer

The short answer (final result) will be obtained at the end of Step 5 after performing polynomial long division and simplification.

Step 1: Rewrite the polynomial division as long division

First, write the given exercise in the form we usually use to perform normal arithmetic division, but this time we will perform a polynomial division.

Step 2: Polynomial Division - First Round

Now divide the first term in the numerator \(2x^{3}\) with the first term in the denominator \(x^{2}\) to get \(2x\). Then, multiply \((x-1)^{2}\) by \(2x\), and subtract the result from the initial polynomial in the numerator. The result of this operation will replace the initial polynomial.

Step 3: Polynomial Division - Second Round

Perform the division again, now with the first term of the new polynomial obtained in Step 2, and the first term of the denominator \((x-1)^{2}\). Remember to subtract the result from the new polynomial, and repeat this process until no terms are left to divide.

Step 4: Simplification

After all terms have been divided, simplify the obtained expression to its lowest terms.

Step 5: Final Answer

The final result obtained is the answer to the long division of the polynomials.