Problem 25 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use synthetic division to divide. $$\left(6 x^{3}+7 x^{2}-x+26\right) \div(x-3)$$

Step-by-Step Solution

Answer

The quotient of the synthetic division is \(6x^{2} + 25x + 74\) and the remainder is 248.

Step 1: Setup Synthetic Division Tableau

First, make sure the polynomial is written in descending order of exponent. Then, write the coefficients (6, 7, -1, 26) in a row. The number to be put into the box is the opposite of the constant term in the divisor, which is \(x-3\) in this case. That constant becomes +3.

Step 2: Carry Out Synthetic Division

Now carry out the synthetic division. Bring down the leading coefficient (6). Multiply 6 by 3 (value from the box), to get 18, and write this value underneath the second coefficient. Add 7 and 18 together to get 25, then multiply 25 by 3 to get 75. Write 75 under -1, giving 74. Multiply 74 by 3 to get 222 and add to 26 yielding 248.

Step 3: Interpret the Result

The numbers on the bottom row of the tableau are the coefficients of the quotient polynomial. From left to right, these represent coefficients for \(x^{2}\), \(x^{1}\), \(x^{0}\), and the remainder, respectively. Hence, the result of the division is \(6x^{2} + 25x + 74\) with remainder 248