Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use synthetic division to divide. $$\left(5 x^{3}+18 x^{2}+7 x-6\right) \div(x+3)$$

Step-by-Step Solution

Answer

\(5x^2 + 3x - 6 + 15/(x+3)\)

Step 1: Set up the synthetic division

First, you need to write down the coefficients of the polynomial you are dividing, which in this case, are 5, 18, 7, and -6. Then, write the number you are dividing by (the opposite of the constant in the binomial) on the left side outside the division bar. The setup should look like this: -3 | 5 18 7 -6.

Step 2: Carry out the synthetic division

Bring down the first coefficient to start, and then multiply it by the number on the outside. Write this product underneath the next coefficient, and add these two numbers together. Write this sum underneath the line. Repeat this process for all of the coefficients. Here is how it looks: -3 | 5 18 7 -6. This becomes: -3 | 5 -15 7 21. Which simplifies to: -3 | 5 3 7 -15. And finally: -3 | 5 3 -6 21. By following this method, it becomes: -3 | 5 3 -6 15. The final line of numbers represents the coefficients of the quotient polynomial, and the remainder.

Step 3: Write the answer

The numbers on the bottom line represent the coefficients (including the remainder) of the result. The first number will be the coefficient of \(x^2\), the second number will be the coefficient of \(x\), the third number will be the constant term, and the fourth number will be the remainder. So, the solution should be: \(5x^2 + 3x - 6 + 15/(x+3)\).