Question

if $g(x) = 21x^2 - 40x - 36$, use synthetic division to find $g(1)$.

Explanation:

Step1: Recall synthetic division for polynomial evaluation

To find \( g(1) \) using synthetic division for the polynomial \( g(x)=21x^{2}-40x - 36 \), we use the coefficients of the polynomial. The coefficients of \( x^{2}\), \( x\), and the constant term are \( 21\), \( - 40\), and \( -36\) respectively. We set up the synthetic division with the root \( r = 1\).

The setup is:
\[

$$\begin{array}{r|rrr} 1 & 21 & -40 & -36\\ & & 21 & -19\\ \hline & 21 & -19 & -55\\ \end{array}$$

\]

Step2: Interpret the result

In synthetic division, the last number in the bottom row is the value of the polynomial at \( x = r\). Here, when \( r=1\), the last number is \( - 55\). So \( g(1)=-55\).

Alternatively, we can also use the direct substitution method to verify: \( g(1)=21\times(1)^{2}-40\times(1)-36=21 - 40 - 36=21-(40 + 36)=21 - 76=-55\).

Answer:

\(-55\)