Question

if $g(x) = 4x^{4} - 23x^{3} + 10x^{2} + 25x$, use synthetic division to find $g(5)$.

Explanation:

Step1: Rewrite the polynomial

First, we rewrite \( g(x) = 4x^{4}-23x^{3}+10x^{2}+25x \) in standard form with all coefficients (including the constant term, which is 0 here). So \( g(x)=4x^{4}-23x^{3}+10x^{2}+25x + 0 \). The coefficients are \( 4,-23,10,25,0 \) and we are using synthetic division with \( x = 5 \).

Step2: Set up synthetic division

We set up the synthetic division as follows:
Bring down the first coefficient (4):
\[

$$\begin{array}{r|rrrrr} 5 & 4 & -23 & 10 & 25 & 0 \\ & & & & & \\ \hline & 4 & & & & \\ \end{array}$$

\]

Step3: Multiply and add (first time)

Multiply the number we brought down (4) by 5: \( 4\times5 = 20 \). Add this to the next coefficient (-23): \( -23 + 20=-3 \).
\[

$$\begin{array}{r|rrrrr} 5 & 4 & -23 & 10 & 25 & 0 \\ & & 20 & & & \\ \hline & 4 & -3 & & & \\ \end{array}$$

\]

Step4: Multiply and add (second time)

Multiply the new number (-3) by 5: \( -3\times5=-15 \). Add this to the next coefficient (10): \( 10+(-15) = -5 \).
\[

$$\begin{array}{r|rrrrr} 5 & 4 & -23 & 10 & 25 & 0 \\ & & 20 & -15 & & \\ \hline & 4 & -3 & -5 & & \\ \end{array}$$

\]

Step5: Multiply and add (third time)

Multiply the new number (-5) by 5: \( -5\times5=-25 \). Add this to the next coefficient (25): \( 25+(-25)=0 \).
\[

$$\begin{array}{r|rrrrr} 5 & 4 & -23 & 10 & 25 & 0 \\ & & 20 & -15 & -25 & \\ \hline & 4 & -3 & -5 & 0 & \\ \end{array}$$

\]

Step6: Multiply and add (fourth time)

Multiply the new number (0) by 5: \( 0\times5 = 0 \). Add this to the last coefficient (0): \( 0 + 0=0 \).
\[

$$\begin{array}{r|rrrrr} 5 & 4 & -23 & 10 & 25 & 0 \\ & & 20 & -15 & -25 & 0 \\ \hline & 4 & -3 & -5 & 0 & 0 \\ \end{array}$$

\]

The last number in the bottom row is the value of \( g(5) \) when using synthetic division (by the Remainder Theorem, the remainder when dividing by \( x - 5 \) is \( g(5) \)).

Answer:

\( 0 \)