Problem 28 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use synthetic division to divide. $$\left(5 x^{3}+6 x+8\right) \div(x+2)$$

Step-by-Step Solution

Answer

The result of the division \((5x^3+6x+8) \div (x+2)\) using synthetic division is \(5x^2 - 10x + 26 - \frac{44}{x+2}\)

Step 1: Setup Synthetic Division

To set up synthetic division, first write the coefficients of the polynomial being divided (the dividend) to the right of a vertical line. The coefficients of the polynomial are 5 (from \(x^3\)), 0 (from \(x^2\)) - as there is no \(x^2\) term in the dividend, hence the coefficient for this term is 0, 6 (from \(x\))/ and 8 (the constant term ). The root of the divisor \(x+2\) is -2. So, the setup will look something like this: \[\begin{{array}}{{r|rrrr}}-2 & 5 & 0 & 6 & 8 \\end{{array}}\]

Step 2: Perform Synthetic Division

Now, bring down the leading coefficient known as '5' as it is to an empty space below the line under the first column. Then multiply this number by the root, -2, write the result under the next column and then add the column downwards. Repeat this step until every column has been filled. \[\begin{{array}}{{r|rrrrr}}-2 & 5 & 0 & 6 & 8 \ & &-10 & 20 & -52\\hline & 5 &-10 & 26 & -44\\end{{array}}\]So, our table will look like this.

Step 3: Interpret the Result

Only the last number is the remainder. The numbers before it are the coefficients of the result of the division. Starting from the coefficient of the value \(x^2\), the result is \(5x^2 - 10x + 26\), and the remainder is -44. Hence, we can say that \((5x^3+6x+8) \div (x+2) = 5x^2 - 10x + 26 - \frac{44}{x+2}\)