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Chapter 3: Problem 22

Divide using synthetic division. $$\left(5 x^{3}-6 x^{2}+3 x+11\right) \div(x-2)$$

Short Answer

Expert verified
After dividing \((5x^3-6x^2+3x+11)\) by \((x-2)\) using synthetic division, the quotient is \(5x^2-7x+17\) and the remainder is 45

Step by step solution

01

Set-up synthetic division

First, we set up the synthetic division. Writing all coefficients of the polynomial in descending degree order, we have 5, -6, 3, and 11. These are the numbers we'll be working with. To the left of these, we write the zero of the divisor (x-2), which is 2.
02

Apply synthetic division

The synthetic division process starts by dropping the leading term (5 in this case) down. Then, multiply this term by the zero we are dividing by (2 in this case), and write the result under the second term from the preceding row (-6), then add those coming to -7. Repeat this operation, multiplying by the zero and adding to the lower row, until all numbers on the top row have been processed.
03

Interpret the result

After carrying out the synthetic division process, the numbers on the bottom row that we obtain represent the coefficients of the result of our division from highest degree to lowest, except for the final number which is the remainder. If our original polynomial is of degree \(n\), then our result will be of degree \(n-1\). In this case, the original polynomial was of degree 3 (cubic) so our result will be of degree 2 (quadratic).

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