Question

use synthetic division to find the result when $3x^3 + 13x^2 + 5x - 21$ is divided by $x + 3$.

Explanation:

Step1: Identify the root

For the divisor \(x + 3\), the root is \(r=-3\) (since \(x+3 = 0\) gives \(x=-3\)). The coefficients of the dividend \(3x^{3}+13x^{2}+5x - 21\) are \(3\), \(13\), \(5\), \(-21\).

Step2: Set up synthetic division

Write the root \(-3\) on the left and the coefficients in a row:

$$
LATEXBLOCK0
$$

Bring down the leading coefficient \(3\):

$$
LATEXBLOCK1
$$

Step3: Multiply and add

Multiply the brought - down number (\(3\)) by the root (\(-3\)): \(3\times(-3)=-9\). Add this to the next coefficient (\(13\)): \(13+(-9) = 4\).

$$
LATEXBLOCK2
$$

Multiply the new number (\(4\)) by the root (\(-3\)): \(4\times(-3)=-12\). Add this to the next coefficient (\(5\)): \(5+(-12)=-7\).

$$
LATEXBLOCK3
$$

Multiply the new number (\(-7\)) by the root (\(-3\)): \(-7\times(-3) = 21\). Add this to the last coefficient (\(-21\)): \(-21 + 21=0\).

$$
LATEXBLOCK4
$$

The last number is the remainder (which is \(0\) here). The other numbers are the coefficients of the quotient polynomial. The degree of the quotient polynomial is one less than the degree of the dividend polynomial. So the quotient polynomial is \(3x^{2}+4x - 7\).

Answer:

The result of the division is \(3x^{2}+4x - 7\) (with a remainder of \(0\)).