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We want to factor the following expression:

x^(4)+9
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: $\newline$ $x^{4}+9$ $\newline$ 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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Domain and range

Full solution

Q. We want to factor the following expression:

\newline

x^{4}+9

\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.

Step $1$ : Determine Factoring Pattern: The expression $x^{4}+9$ is a sum of two terms, where one term is $x$ raised to the fourth power and the other term is a constant $9$ . We need to determine if there is a factoring pattern that applies to this expression.
Step $2$ : Evaluate Option (A): Option (A) suggests using the square of a binomial pattern, either $(U+V)^{2}$ or $(U-V)^{2}$ . However, these patterns expand to $U^{2} + 2UV + V^{2}$ and $U^{2} - 2UV + V^{2}$ , respectively. Neither of these patterns match the given expression $x^{4}+9$ because there is no middle term in the expression $x^{4}+9$ .
Step $3$ : Evaluate Option (B): Option (B) suggests using the difference of squares pattern, $(U+V)(U-V)$ , which expands to $U^2 - V^2$ . This pattern also does not apply because we have a sum, not a difference, and the expression $x^{4}+9$ does not represent a difference of squares.
Step $4$ : Evaluate Option (C): Option (C) states that we can't use any of the patterns. Since neither the square of a binomial pattern nor the difference of squares pattern applies to the expression $x^{4}+9$ , and there are no other common factoring patterns for a sum of a fourth power and a constant, we conclude that the expression $x^{4}+9$ cannot be factored using standard algebraic patterns over the real numbers.

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