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 by \(x - 5\), we use \(r = 5\) (from \(x - r\)). The coefficients of the dividend \(3x^{3}-10x^{2}-21x - 20\) are \(3\), \(-10\), \(-21\), \(-20\).
Set up the synthetic division as:

$$
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$$

Step2: Bring down the leading coefficient

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

$$
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$$

Step3: Multiply and add

Multiply \(3\) by \(5\) to get \(15\), then add to \(-10\): \(-10 + 15=5\).

$$
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$$

Multiply \(5\) by \(5\) to get \(25\), then add to \(-21\): \(-21+25 = 4\).

$$
LATEXBLOCK3
$$

Multiply \(4\) by \(5\) to get \(20\), then add to \(-20\): \(-20 + 20=0\).

$$
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$$

Step4: Interpret the result

The bottom row gives the coefficients of the quotient polynomial and the remainder. The remainder is \(0\), and the quotient is a quadratic polynomial with coefficients \(3\), \(5\), \(4\), so the quotient is \(3x^{2}+5x + 4\).

Answer:

The result is \(3x^{2}+5x + 4\) (with a remainder of \(0\)).