Problem 21 - College Algebra

Question

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

Step-by-Step Solution

Answer

The result of the division is \(4x^2 - 7x + 10 - \frac{11}{x-1}\)

Step 1: Write down the coefficients

The first step is to write down the coefficients of the polynomial in descending order of the powers of \(x\). For the polynomial \(4x^3-3x^2+3x-1\), the coefficients are \[4, -3, 3, -1\]

Step 2: Write down the value of the divisor

For the divisor \(x - 1\), write it as \(-1\). This is because in synthetic division we take the number which, when substituted into the divisor, would give zero. That is for \(x - 1 = 0\), \(x = 1\)

Step 3: Synthetic Division process

The actual process of Synthetic Division is now carried out. The rules are: a) 'Drop down' the first coefficient (in this case \[4\]) b) Multiply the value just calculated by the divisor and put this below the next coefficient (in this case, \[4*(-1) = -4\] goes under -3) c) Add the column to get the next number (in this case, \[-3 + (-4) = -7\] d) Repeat steps b & c until the end. Following this process, we get the values \[4, -7, 10, -11\]. The last number, -11, is the remainder

Step 4: Write down the final answer

The last step is to write down the answer. The values obtained are the coefficients of the polynomial quotient and the remainder. The powers of \(x\) are one less than in the original polynomial. The final answer is thus \[4x^2 - 7x + 10 - \frac{11}{x-1}\](the '-11' is the remainder and put in the form of a fraction with the divisor)