Answer:
[tex]a = -9[/tex].
Step-by-step explanation:
Let [tex]b[/tex] be a real number. Consider the factor theorem: if [tex](x - b)[/tex] is a factor of the function [tex]f(x)[/tex], then it must be true that [tex]f(b) = 0[/tex].
To solve this question, assume that [tex]a[/tex] has already been found. Since [tex](x - 3)[/tex] is a factor of this polynomial, [tex]f(3) = 0[/tex] by the factor theorem.
The left-hand side of this equation can be expressed as:
[tex]\begin{aligned}f(3) &= 3^3 + 3^2 + 3\, a - 9 = 27 + 3\, a\end{aligned}[/tex].
That should match the [tex]0[/tex] on the right-hand side. In other words:
[tex]27 + 3\, a= 0[/tex].
Solve for [tex]a[/tex]:
[tex]\displaystyle a = -\frac{27}{3} = -9[/tex].
