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We want to factor the following expression:

x^(4)+3x^(2)y+9y^(2)
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: $\newline$ $x^{4}+3 x^{2} y+9 y^{2}$ $\newline$ 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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Compare linear and exponential growth

Full solution

Q. We want to factor the following expression:

\newline

x^{4}+3 x^{2} y+9 y^{2}

\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.

Rephrasing the given expression: Let's first rephrase the ＂Which factoring pattern can be used to factor the expression $x^4 + 3x^2y + 9y^2$ ?＂
Identifying the structure of the expression: Identify the structure of the given expression: $x^4 + 3x^2y + 9y^2$ . Notice that $x^4$ is a square $(x^2)^2$ , and $9y^2$ is also a square $(3y)^2$ . The middle term, $3x^2y$ , is twice the product of $x^2$ and $3y$ . This suggests that the expression might be a perfect square trinomial.
Recalling the pattern for a perfect square trinomial: Recall the pattern for a perfect square trinomial: $(U + V)^2 = U^2 + 2UV + V^2$ or $(U - V)^2 = U^2 - 2UV + V^2$ . We need to determine if the given expression fits either of these patterns.
Comparing the expression with the perfect square trinomial patterns: Compare the given expression with the perfect square trinomial patterns. We have $U^2 = x^4$ (so $U = x^2$ ), $V^2 = 9y^2$ (so $V = 3y$ ), and the middle term $2UV$ should be $2 \times x^2 \times 3y = 6x^2y$ . However, the middle term in our expression is $3x^2y$ , not $6x^2y$ . This means that the expression does not fit the perfect square trinomial pattern exactly.
Considering the other factoring pattern: Since the expression does not fit the perfect square trinomial pattern, let's consider the other factoring pattern: $(U + V)(U - V) = U^2 - V^2$ . This pattern is used for the difference of squares, but our expression is not a difference, it's a sum. Therefore, this pattern does not apply either.
Concluding that the expression cannot be factored: Having considered the common factoring patterns and finding that none of them apply to the given expression, we conclude that the expression $x^4 + 3x^2y + 9y^2$ cannot be factored using the patterns $(U + V)^2$ , $(U - V)^2$ , or $(U + V)(U - V)$ .

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