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Use the remainder theorem to complete the informal proof for the following statement.

[tex]$x = -5$[/tex] is a root of the function [tex]$f(x) = x^3 + 3x^2 - 25x - 75$[/tex].

1. If [tex]$x = -5$[/tex] is a root of [tex]$f(x)$[/tex], then [tex]$(x + 5)$[/tex] must be _____.
2. Find the [tex]$\square$[/tex] of [tex]$f(x)$[/tex] and [tex]$(x + 5)$[/tex].
3. The [tex]$\square$[/tex] of this operation is 0.
4. Thus, [tex]$(x + 5)$[/tex] is _____ [tex]$f(x)$[/tex].
5. Therefore, [tex]$x = -5$[/tex] is a root of [tex]$f(x)$[/tex].



Answer :

GinnyAnswer
Sure, let's use the remainder theorem and the given answer to complete this proof step-by-step.

1. If [tex]\(x = -5\)[/tex] is a root of [tex]\(f(x)\)[/tex], then [tex]\((x + 5)\)[/tex] must be a factor of [tex]\(f(x)\)[/tex].

2. Find the remainder of [tex]\(f(x)\)[/tex] when divided by [tex]\((x + 5)\)[/tex].

3. The result of this operation is 0.

4. Thus, [tex]\((x + 5)\)[/tex] is a factor of [tex]\(f(x)\)[/tex].

5. Therefore, [tex]\(x = -5\)[/tex] is a root of [tex]\(f(x)\)[/tex].