Sure, let's go through the solution to see if [tex]\( x = 12 \)[/tex] is a zero of the polynomial [tex]\( n(x) = 3x^3 - x^2 - 39x + 13 \)[/tex] by using synthetic division and the remainder theorem.
### Synthetic Division Steps:
The polynomial is [tex]\( n(x) = 3x^3 - x^2 - 39x + 13 \)[/tex].
Given [tex]\( x = 12 \)[/tex]:
1. Write down the coefficients:
The coefficients of [tex]\( n(x) \)[/tex] are [tex]\( [3, -1, -39, 13] \)[/tex].
2. Perform synthetic division:
- Start with the leading coefficient [tex]\( 3 \)[/tex].
- Multiply by [tex]\( x = 12 \)[/tex] and add the result to the next coefficient.
Let's perform the steps:
1. Bring down the first coefficient (3):
[tex]\[
3
\][/tex]
2. Multiply by [tex]\( 12 \)[/tex] and add to the second coefficient [tex]\( -1 \)[/tex]:
[tex]\[
3 \cdot 12 = 36
\][/tex]
[tex]\[
36 + (-1) = 35
\][/tex]
3. Multiply the result by [tex]\( 12 \)[/tex] and add to the third coefficient [tex]\( -39 \)[/tex]:
[tex]\[
35 \cdot 12 = 420
\][/tex]
[tex]\[
420 + (-39) = 381
\][/tex]
4. Multiply the result by [tex]\( 12 \)[/tex] and add to the constant term [tex]\( 13 \)[/tex]:
[tex]\[
381 \cdot 12 = 4572
\][/tex]
[tex]\[
4572 + 13 = 4585
\][/tex]
### Remainder Result:
Based on the synthetic division, the remainder is [tex]\( 4585 \)[/tex].
The Remainder Theorem states that if you substitute a number [tex]\( x = c \)[/tex] into a polynomial [tex]\( n(x) \)[/tex] and the result is 0, then [tex]\( x = c \)[/tex] is a zero of the polynomial.
In this case, substituting [tex]\( x = 12 \)[/tex] into the polynomial yielded the remainder [tex]\( 4585 \)[/tex], which is not zero.
### Conclusion:
- The remainder is [tex]\( 4585 \)[/tex].
- Since the remainder is not zero, [tex]\( x = 12 \)[/tex] is not a zero of the polynomial [tex]\( n(x) \)[/tex].
