Problem 25 - College Algebra

Question

Divide using synthetic division. $$\left(x^{2}-5 x-5 x^{3}+x^{4}\right) \div(5+x)$$

Step-by-Step Solution

Answer

The result of the division is \(x^{3} + 5x^{2} + 25x + 125\).

Step 1: Arrange the Polynomial

First, arrange the polynomial in decreasing level of exponents and if any exponents are missing, a term with 0 should be added for the missing exponents. For this problem, the polynomial arranged correctly would be \(x^{4} - 5x^{3} -5x^{2} + 0x +0\).

Step 2: Set Up Synthetic Division

Set up the synthetic division by writing the coefficients of the polynomial (1, -5, -5, 0, 0) in a row. Since we are dividing by a linear expression \(5 + x\), set up the synthetic division with -5 (as synthetic division specifically deals with roots of the divisor). The coefficient from our roots is very important to be placed correctly which would be -5.

Step 3: Perform Synthetic Division

The first step in synthetic division is to bring down the first coefficient (in this case 1). Then you multiply this number by the root (in this case -5), and write the result under the second coefficient and add the column, writing the result beneath. Continue this process for all the coefficients. For this problem you should end up with a row of coefficients (1, 0, 5, 25, 125). The last number 125 is the remainder.

Step 4: Write out the Quotient

Finally, these coefficients represent the coefficients for the quotient polynomial, starting with an exponent one lower than the original polynomial. The final quotient, therefore, is \(x^{3} + 5x^{2} + 25x + 125\).