Question

question
use synthetic division to find the result when $3x^3 - 10x^2 - 21x - 20$ is divided by $x - 5$.

Explanation:

Step1: Set up synthetic division

For dividing \(3x^{3}-10x^{2}-21x - 20\) by \(x - 5\), we use the root of the divisor \(x-5 = 0\), so the root is \(x = 5\). The coefficients of the dividend are \(3\), \(- 10\), \(-21\), \(-20\).
Set up the synthetic division as:

$$
LATEXBLOCK0
$$

Step2: Bring down the leading coefficient

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

$$
LATEXBLOCK1
$$

Step3: Multiply and add

Multiply the number we just brought down (\(3\)) by the root (\(5\)): \(3\times5=15\). Add this to the next coefficient (\(-10\)): \(-10 + 15=5\). Write this result in the bottom row:

$$
LATEXBLOCK2
$$

Step4: Repeat the multiply - add process

Multiply the new number in the bottom row (\(5\)) by the root (\(5\)): \(5\times5 = 25\). Add this to the next coefficient (\(-21\)): \(-21+25 = 4\). Write this result in the bottom row:

$$
LATEXBLOCK3
$$

Step5: Repeat the multiply - add process again

Multiply the new number in the bottom row (\(4\)) by the root (\(5\)): \(4\times5=20\). Add this to the last coefficient (\(-20\)): \(-20 + 20=0\). Write this result in the bottom row:

$$
LATEXBLOCK4
$$

The bottom row (excluding the last number which is the remainder) gives the coefficients of the quotient polynomial. Since the original polynomial is of degree \(3\), the quotient polynomial will be of degree \(2\). The coefficients are \(3\), \(5\), \(4\), so the quotient is \(3x^{2}+5x + 4\) and the remainder is \(0\).

Answer:

The result of the division is \(3x^{2}+5x + 4\) (with a remainder of \(0\)). So when \(3x^{3}-10x^{2}-21x - 20\) is divided by \(x - 5\), the quotient is \(3x^{2}+5x + 4\) and the remainder is \(0\), and we can write \(3x^{3}-10x^{2}-21x - 20=(x - 5)(3x^{2}+5x + 4)\)