To determine if [tex]\( x = 1 \)[/tex] is a zero of the polynomial [tex]\( n(x)=3x^3-x^2-39x+13 \)[/tex], we need to use the Remainder Theorem, which states that if a polynomial [tex]\( p(x) \)[/tex] is divided by [tex]\( x-a \)[/tex], the remainder of this division is [tex]\( p(a) \)[/tex]. Thus, we evaluate [tex]\( n(1) \)[/tex].
We use synthetic division for this:
1. Begin with the coefficients of the polynomial: [tex]\( 3, -1, -39, 13 \)[/tex].
[tex]\[
\begin{array}{r|rrrr}
1 & 3 & -1 & -39 & 13 \\
\hline
& 3 & 2 & -37 & -24 \\
\end{array}
\][/tex]
- Bring down the leading coefficient (3):
- [tex]\( 3 \)[/tex]
- Multiply 3 by 1 (the value being tested):
- [tex]\( 3 \cdot 1 = 3 \)[/tex]
- Add this product to the next coefficient:
- [tex]\( -1 + 3 = 2 \)[/tex]
- Multiply 2 by 1:
- [tex]\( 2 \cdot 1 = 2 \)[/tex]
- Add this product to the next coefficient:
- [tex]\( -39 + 2 = -37 \)[/tex]
- Multiply -37 by 1:
- [tex]\( -37 \cdot 1 = -37 \)[/tex]
- Add this product to the next coefficient:
- [tex]\( 13 + (-37) = -24 \)[/tex]
The remainder when [tex]\( x = 1 \)[/tex] is tested is:
[tex]\[
\boxed{-24}
\][/tex]
Since the remainder is not zero, [tex]\( x = 1 \)[/tex] is not a zero of the polynomial [tex]\( n(x) \)[/tex].
