Problem 23 - College Algebra

Question

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

Step-by-Step Solution

Answer

The solution is \(6x^4 + 10x^3 + 20x^2 + 40x + 77 + \frac{155}{x-2}\)

Step 1: Arrange the synthetic division set up

First, the coefficients from the polynomial \(6 x^{5}-2 x^{3}+4 x^{2}-3 x+1\) are arranged over a synthetic division bar. The number from the divisor \(x-2\) (which is 2) is placed outside the L-shape bar. Include 0s for any missing terms.

Step 2: Synthetic Division Process

Bring down the first coefficient (6) which lands under the solution bar. Multiply this by 2 (the number outside the L-shape) to get 12. Add this to the next coefficient (-2) to get 10, which also places under the solution bar. Repeat this process until you reach the last coefficient.

Step 3: Express the result as a polynomial

The numbers under the solution bar become the coefficients of the answer polynomial. As we've used a 5th degree polynomial, the answer polynomial starts with degree 4 (one less). Write these coefficients with corresponding variables with decreasing powers. The last number under the solution bar is the remainder from division, and should be written as a fraction over the divisor. If there's no remainder, this step can be omitted.