Question

find the quotient and remainder using synthetic division: $\frac{x^{3}-23}{x-3}$
the quotient is
the remainder is
question help: video read written example

Explanation:

Step1: Set up synthetic division

Divisor root: $x=3$. Coefficients of dividend: $1, 0, 0, -23$ (for $x^3+0x^2+0x-23$)

Step2: Bring down leading coefficient

$$\begin{array}{r|rrrr}3&1&0&0&-23\\&&&&\\\hline&1&&&\\\end{array}$$

Step3: Multiply and add (column1)

Multiply $1\times3=3$, add to next coefficient: $0+3=3$

$$\begin{array}{r|rrrr}3&1&0&0&-23\\&&3&&\\\hline&1&3&&\\\end{array}$$

Step4: Multiply and add (column2)

Multiply $3\times3=9$, add to next coefficient: $0+9=9$

$$\begin{array}{r|rrrr}3&1&0&0&-23\\&&3&9&\\\hline&1&3&9&\\\end{array}$$

Step5: Multiply and add (column3)

Multiply $9\times3=27$, add to last coefficient: $-23+27=4$

$$\begin{array}{r|rrrr}3&1&0&0&-23\\&&3&9&27\\\hline&1&3&9&4\\\end{array}$$

Step6: Identify quotient and remainder

Quotient coefficients: $1,3,9$ → $x^2+3x+9$; Remainder: $4$

Answer:

The quotient is $x^2+3x+9$
The remainder is $4$