Question

Use synthetic division to find the result when 2, x, cubed, minus, 8, x, squared, minus, 19, x, minus, 302x
3
−8x
2
−19x−30 is divided by x, minus, 6x−6.

Answer 1

To perform synthetic division with the polynomial \(2x^3 - 8x^2 - 19x - 30\) by the binomial \(x - 6\), we first need to use the zero of the divisor, which is \(x = 6\).

1. Write down the coefficients of the polynomial:
- The coefficients for \(2x^3 - 8x^2 - 19x - 30\) are \(2, -8, -19, -30\).

2. Set up synthetic division:
\[
\begin{array}{r|rrrr}
6 & 2 & -8 & -19 & -30 \\
& & 12 & 24 & 30 \\
\hline
& 2 & 4 & 5 & 0 \\
\end{array}
\]

3. Perform synthetic division steps:
- Bring down the first coefficient: \(2\).
- Multiply \(6\) by \(2\) (the value below the line) to get \(12\) and add it to \(-8\):
\(-8 + 12 = 4\).
- Next, multiply \(6\) by \(4\) to get \(24\) and add it to \(-19\):
\(-19 + 24 = 5\).
- Now multiply \(6\) by \(5\) to get \(30\) and add it to \(-30\):
\(-30 + 30 = 0\).

4. The result of the synthetic division gives us:
\[
2x^2 + 4x + 5
\]
with a remainder of \(0\).

Thus, when \(2x^3 - 8x^2 - 19x - 30\) is divided by \(x - 6\), the result is:
\[
2x^2 + 4x + 5
\]