Sure, let's use polynomial long division to determine which of the given expressions is not a factor of the polynomial [tex]\(3x^3 - 10x^2 - 143x - 90\)[/tex]. We'll divide the polynomial by each of the given expressions and check for a remainder. If the remainder is zero, then the expression is a factor; otherwise, it is not.
1. Division by [tex]\(x - 6\)[/tex]:
First, we set up the division of [tex]\(3x^3 - 10x^2 - 143x - 90\)[/tex] by [tex]\(x - 6\)[/tex].
[tex]\[
\begin{array}{r|rrr}
& 3x^2 + 8x + 5 & \\
\hline
x-6 & 3x^3 - 10x^2 - 143x - 90 \\
& -(3x^3 - 18x^2) \\
\hline
& 8x^2 - 143x \\
& -(8x^2 - 48x) \\
\hline
& -95x - 90 \\
& -(- 95x + 570) \\
\hline
& -660 \\
\end{array}
\][/tex]
The remainder is [tex]\(-660\)[/tex]. Since the remainder is not zero, [tex]\(x - 6\)[/tex] is not a factor.
2. Division by [tex]\(x - 9\)[/tex]:
Next, we set up the division of [tex]\(3x^3 - 10x^2 - 143x - 90\)[/tex] by [tex]\(x - 9\)[/tex].
[tex]\[
\begin{array}{r|rrr}
& 3x^2 + 17x + 12 & \\
\hline
x-9 & 3x^3 - 10x^2 - 143x - 90 \\
& -(3x^3 - 27x^2) \\
\hline
& 17x^2 - 143x \\
& -(17x^2 - 153x) \\
\hline
& -296x - 90 \\
& -(- 296x + 2670) \\
\hline
& -90 \\
\end{array}
\][/tex]
The remainder is 0. Since the remainder is zero, [tex]\(x - 9\)[/tex] is a factor.
3. Division by [tex]\(3x + 2\)[/tex]:
Next, we set up the division of [tex]\(3x^3 - 10x^2 - 143x - 90\)[/tex] by [tex]\(3x + 2\)[/tex].
[tex]\[
\begin{array}{r|rrr}
& x^2 - 4x - 45 & \\
\hline
3x+2 & 3x^3 - 10x^2 - 143x - 90 \\
& -(3x^3 + 2x^2) \\
\hline
& -12x^2 - 143x \\
& -(-12x^2 - 8x) \\
\hline
& -151x - 90 \\
& -(- 151x - 22) \\
\hline
& 0 \\
\end{array}
\][/tex]
The remainder is 0. Since the remainder is zero, [tex]\(3x + 2\)[/tex] is a factor.
4. Division by [tex]\(x + 5\)[/tex]:
Finally, we set up the division of [tex]\(3x^3 - 10x^2 - 143x - 90\)[/tex] by [tex]\(x + 5\)[/tex].
[tex]\[
\begin{array}{r|rrr}
& 3x^2 - 25x - 18 & \\
\hline
x+5 & 3x^3 - 10x^2 - 143x - 90 \\
& -(3x^3 + 15x^2) \\
\hline
& -25x^2 - 143x \\
& -(-25x^2 - 125x) \\
\hline
& -18x - 90 \\
& -(- 18x - 90) \\
\hline
& 0 \\
\end{array}
\][/tex]
The remainder is 0. Since the remainder is zero, [tex]\(x + 5\)[/tex] is a factor.
Based on our calculations, [tex]\(x - 6\)[/tex] is not a factor of the polynomial [tex]\(3x^3 - 10x^2 - 143x - 90\)[/tex].
