Answer:
[tex]A = 6[/tex].
Step-by-step explanation:
Let [tex]a[/tex] denote a constant (a number, rather than a variable like [tex]x[/tex].)
Let [tex]f(x)[/tex] denote an expression about [tex]x[/tex] (for example,
By the factor theorem, if [tex](x - a)[/tex] is a factor of [tex]f(x)[/tex], then [tex]f(a) = 0[/tex]. In other words, if all mentions of [tex]x[/tex] in the expression [tex]f(x)\![/tex] are replaced with [tex]a[/tex], then the expression should evaluate to zero.
[tex](x + 3)[/tex] is equivalent to [tex](x - (-3))[/tex]. That is: [tex]a = -3[/tex].
Replace all mentions of [tex]x[/tex] in [tex]x^3 - x^2 - 10\, x + A[/tex] by [tex](-3)[/tex] to get:
[tex](-3)^3 - (-3)^2 - 10\times (-3) + A[/tex].
That should evaluate to zero. Therefore: [tex]-27 - 9 + 30 + A = 0[/tex]. Solve for [tex]A[/tex] to get [tex]A = 6[/tex].
