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Chapter 3: Problem 21

Divide using synthetic division. $$\left(4 x^{3}-3 x^{2}+3 x-1\right) \div(x-1)$$

Short Answer

Expert verified
The division \((4x^3-3x^2+3x-1) \div (x-1)\) using synthetic division results in \(4x^2+1x+4\) remainder 3.

Step by step solution

01

Set up synthetic division

First, the coefficients of the cubic polynomial, in the order of highest to lowest degree, are written in a row: 4, -3, 3, -1. The zero of the divisor is identified by setting the linear divisor equal to zero and solving for x, giving x=1. This zero is written to the left of the coefficient row.
02

Perform synthetic division

Next, the first number (4) is brought down under the line. Each subsequent number under the line is found by multiplying the zero (1) by the number above currently, and summing this product with the next number to its right from the initial coefficient row. First, \(1*4 = 4\), added to -3 gives 1, so 1 is written under the -3. Then \(1*1=1\), added to 3 gives 4, so 4 is written under the 3. Finally, \(1*4=4\), added to -1 gives 3, so 3 is written under the -1.
03

Write the final answer

Finally, the answer is written out by forming a polynomial from the numbers under the line. The degree of the first term is one less than the degree of the original polynomial. So the numbers form a quadratic polynomial with coefficients 4, 1, 4 and the remainder is 3, so \(4x^2+1x+4\) and remainder 3.

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