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We want to factor the following expression:

9x^(6)+6x^(3)y+y^(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: $\newline$ $9 x^{6}+6 x^{3} y+y^{2}$ $\newline$ 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

9 x^{6}+6 x^{3} y+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.

Recognize the structure: Recognize the structure of the given expression. $\newline$ The given expression is $9x^{6} + 6x^{3}y + y^{2}$ . We can observe that the first term is a perfect square $(3x^{3})^2$ , the last term is a perfect square $(y)^2$ , and the middle term is twice the product of the square roots of the first and last terms $(2 \times 3x^{3} \times y)$ .
Identify the factoring pattern: Identify the factoring pattern. $\newline$ The structure of the expression matches the pattern of a perfect square trinomial, which is $(A + B)^2 = A^2 + 2AB + B^2$ or $(A - B)^2 = A^2 - 2AB + B^2$ . In this case, since the middle term is positive, we will use the $(A + B)^2$ pattern.
Apply the factoring pattern: Apply the factoring pattern to the expression. $\newline$ We can write the expression as $(3x^{3} + y)^2$ because $(3x^{3})^2 = 9x^{6}$ , $2 \cdot 3x^{3} \cdot y = 6x^{3}y$ , and $y^2 = y^{2}$ . Therefore, the factored form of the expression is $(3x^{3} + y)^2$ .

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