Question

use synthetic division to find the result when $3x^3 - 6x^2 - 14x + 15$ is divided by $x - 3$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.

Explanation:

Step1: Identify the root for synthetic division

For the divisor \(x - 3\), the root \(c\) is \(3\) (since \(x - 3=0\) gives \(x = 3\)). The coefficients of the dividend \(3x^{3}-6x^{2}-14x + 15\) are \(3\), \(-6\), \(-14\), and \(15\).

Step2: Set up synthetic division

Write the root \(3\) to the left, and the coefficients in a row:

$$
LATEXBLOCK0
$$

Step3: Bring down the leading coefficient

Bring down the \(3\) to the bottom row:

$$
LATEXBLOCK1
$$

Step4: Multiply and add

Multiply the root \(3\) by the number just brought down (\(3\)): \(3\times3 = 9\). Add this to the next coefficient (\(-6\)): \(-6+9 = 3\). Write this result in the bottom row:

$$
LATEXBLOCK2
$$

Step5: Repeat the multiply - add process

Multiply \(3\) (the root) by the new number in the bottom row (\(3\)): \(3\times3=9\). Add this to the next coefficient (\(-14\)): \(-14 + 9=-5\). Write this in the bottom row:

$$
LATEXBLOCK3
$$

Step6: Repeat again for the last coefficient

Multiply \(3\) by \(-5\): \(3\times(-5)=-15\). Add this to the last coefficient (\(15\)): \(15+(-15) = 0\). Write this in the bottom row:

$$
LATEXBLOCK4
$$

Step7: Interpret the results

The bottom row gives the coefficients of the quotient polynomial and the remainder. The degree of the quotient polynomial is one less than the degree of the dividend. Since the dividend is a cubic (\(3x^{3}\)), the quotient is a quadratic. The coefficients are \(3\) (for \(x^{2}\)), \(3\) (for \(x\)), and \(-5\) (constant term), and the remainder is \(0\). So the quotient \(q(x)=3x^{2}+3x - 5\) and the remainder \(r(x) = 0\).

Since the remainder is \(0\), the division result is \(3x^{2}+3x - 5+\frac{0}{x - 3}\), which simplifies to \(3x^{2}+3x - 5\).

Answer:

\(3x^{2}+3x - 5\)