Question

question
use synthetic division to find the result when $2x^3 - 3x^2 + 5x - 14$ is divided by $x - 2$.

answer attempt 1 out of 2

Explanation:

Step1: Set up synthetic division

For dividing \(2x^3 - 3x^2 + 5x - 14\) by \(x - 2\), we use the root of \(x - 2 = 0\), which is \(x = 2\). The coefficients of the polynomial are \(2\), \(-3\), \(5\), \(-14\).
Set up the synthetic division as:

$$
LATEXBLOCK0
$$

Step2: Bring down the leading coefficient

Bring down the \(2\) to the bottom row:

$$
LATEXBLOCK1
$$

Step3: Multiply and add

Multiply the \(2\) (bottom row) by \(2\) (root) to get \(4\), then add to \(-3\): \(-3 + 4 = 1\).

$$
LATEXBLOCK2
$$

Multiply \(1\) (bottom row) by \(2\) to get \(2\), add to \(5\): \(5 + 2 = 7\).

$$
LATEXBLOCK3
$$

Multiply \(7\) (bottom row) by \(2\) to get \(14\), add to \(-14\): \(-14 + 14 = 0\).

$$
LATEXBLOCK4
$$

Step4: Write the result

The bottom row coefficients \(2\), \(1\), \(7\) with a remainder of \(0\) mean the quotient is a quadratic polynomial \(2x^2 + x + 7\) (since the degree of the quotient is one less than the dividend, and remainder is \(0\)).

Answer:

\(2x^2 + x + 7\)