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Chapter 3: Problem 18

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

Short Answer

Expert verified
The quotient of the given polynomial division is \(x + 2\) with remainder 0.

Step by step solution

01

Setting up the Synthetic Division

Firstly, you need to write down coefficients of the polynomial i.e., \(x^2 + x - 2\). Here, coefficients are 1, 1, and -2. Afterwards, write down the zero of divisor \(x - 1\). The zero can be found by setting \(x - 1 = 0\), so \(x = 1)$. Set up your synthetic division as such: \[\begin{array}{c|ccc} 1 & 1 & 1 & -2 \end{array}\].
02

Performing Synthetic Division

Bring down the leading coefficient (which is 1 in this case) to the bottom row. Afterwards, multiply this number by 1 (the zero of divisor) and write the product under the next coefficient. The row should look like this: \[\begin{array}{c|cc|c} 1 & 1 & 1 & -2 \ & & 1 & \end{array}\]. Add the numbers in the column, write the sum at the bottom and repeat this process for the rest of the coefficients until the last column where you will get the remainder. The final table should look like this: \[\begin{array}{c|cc|c} 1 & 1 & 1 & -2 \ & 1 & 2 & \end{array}\] \[\begin{array}{c|c|c|c} 1 & 2 & 2 & 0 \end{array}\].
03

Writing the Quotient

Finally, interpret your answer, the numbers in the last row represent the coefficients of quotient. Start from one degree lower than the original polynomial and continue till constant term. Here, the quotient will be \(x + 2\) and the remainder is 0.

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