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We want to factor the following expression:
x^(3)-25
Which pattern can we use to factor the expression?
U and V are either constant integers or single-variable expressions.
Choose 1 answer:
(A) (U+V)^(2) or (U-V)^(2)
(B) (U+V)(U-V)
(C) We can't use any of the patterns.

We want to factor the following expression: $\newline$ $x^{3}-25$ $\newline$ Which pattern can we use to factor the expression? $\newline$ $U$ and $V$ are either constant integers or single-variable expressions. $\newline$ Choose $1$ answer: $\newline$ (A) $(U+V)^{2}$ or $(U-V)^{2}$ $\newline$ (B) $(U+V)(U-V)$ $\newline$ (C) We can't use any of the patterns.

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Complex conjugate theorem

Full solution

Q. We want to factor the following expression:

\newline

x^{3}-25

\newline

Which pattern can we use to factor the expression?

\newline

U

and

V

are either constant integers or single-variable expressions.

\newline

Choose

1

answer:

\newline

(A)

(U+V)^{2}

or

(U-V)^{2}

\newline

(B)

(U+V)(U-V)

\newline

(C) We can't use any of the patterns.

Identify expression: Identify the expression to be factored. $\newline$ The expression given is $x^3 - 25$ . We need to find a pattern to factor this expression.
Recognize pattern: Recognize the pattern applicable for the expression. $\newline$ The expression $x^3 - 25$ can be seen as a difference of cubes, where $x^3$ is a cube and $25$ can be written as $5^3$ . Therefore, the expression can be rewritten as $x^3 - 5^3$ . The pattern for factoring a difference of cubes is $(U^3 - V^3) = (U - V)(U^2 + UV + V^2)$ , where $U$ and $V$ are either constant integers or single-variable expressions.
Apply pattern: Apply the pattern to factor the expression. $\newline$ Using the pattern for the difference of cubes, we can factor $x^3 - 5^3$ as follows: $\newline$ Let $U = x$ and $V = 5$ , then $\newline$ $(x^3 - 5^3) = (x - 5)(x^2 + 5x + 25)$ .

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