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Chapter 3: Problem 23

Divide using synthetic division. $$\left(6 x^{5}-2 x^{3}+4 x^{2}-3 x+1\right) \div(x-2)$$

Short Answer

Expert verified
\((6x^{5} - 2x^{3} + 4x^{2} - 3x + 1)\) divided by \((x - 2)\) equals \(6x^4 + 12x^3 + 22x^2 + 44x + 85 + \frac{171}{x-2}\)

Step by step solution

01

Set Up the Synthetic Division

First, learn what the synthetic division set-up looks like. Here, it is done by taking the coefficients of the dividend \((6, 0, -2, 4, -3, 1)\) and the root of the divisor (which is the solution of the equation \(x - 2 = 0)\), in this case it's \(2\). If there is a missing term in the polynomial, a zero is used in place of the missing term's coefficient. Thus, the coefficient for \(x^4\) is \(0\). The next step is to bring down the leading coefficient (which is 6 in this case) to the bottom bar.
02

Multiply and Add

Next, you'll multiply this leading coefficient with the root (which is \(2\)) and place it in line with the coefficient of the next term (which is \(0\)). Then you add together \(0\) and \((6*2)\) resulting in \(12\). You'll continue this process of multiplying and adding with the rest of the coefficients.
03

Write Out the Result

The numbers that are now on the bottom row represent the coefficients of the quotient polynomial. The terms will begin one degree less than the original polynomial, in this case starting with \(x^4\). So, the solution to the problem, or the result of the synthetic division, is \(6x^4 + 12x^3 + 22x^2 + 44x + 85 + \frac{171}{x-2}\). In other words, \((6x^{5} - 2x^{3} + 4x^{2} - 3x + 1)\) divided by \((x - 2)\) equals \(6x^4 + 12x^3 + 22x^2 + 44x + 85 + \frac{171}{x-2}\)

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