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Chapter 2: Problem 27

Use synthetic division to divide. \(\left(3 x^{3}-17 x^{2}+15 x-25\right) \div(x-5)\)

Short Answer

Expert verified
The quotient polynomial is \(3x^{2} - 2x + 5 - R\), where \(R\) is the remainder of the division.

Step by step solution

01

Identifying Coefficients

First, write down the coefficients of the cubic polynomial in order. This gives: \(3, -17, 15, -25\). Then, identify the constant value from \(x-5\) which is -5. But we need to take it as positive 5, because we subtract it in the equation.
02

Setting up Synthetic Division

Draw a 'L' shape to start the synthetic division. Write the numbers identified from the previous step outside the 'L', beginning from the top: 3, -17, 15, -25 from top to bottom. Then, bring down the first number, \(3\), to the bottom of the 'L'.
03

Performing Synthetic Division

Multiply the number brought down with the constant 5 (from \(x-5\)) and then put the result up. Then add the column down and multiply with the constant from the binomial again. Put result in the next column and repeat this process for all columns. Proceed till you reach the end. The result are the coefficients of the quotient polynomial, and the remainder of the division.
04

Writing Quotient Polynomial

Finally, write down the quotient polynomial using the coefficients obtained from the synthetic division. The coefficients represent the terms in the polynomial from highest power down to the constant term, so the last term represents the remainder.

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