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We want to factor the following expression:

(x-3)^(2)-10(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: $\newline$ $(x-3)^{2}-10(x-3)+25$ $\newline$ 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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Compare linear and exponential growth

Full solution

Q. We want to factor the following expression:

\newline

(x-3)^{2}-10(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.

Expression Analysis: We are given the expression $(x-3)^{2}-10(x-3)+25$ and asked to identify a factoring pattern that can be used to factor it. Let's first rewrite the expression to see if it resembles any common factoring patterns.
Perfect Square Trinomial: The given expression is a quadratic in the form of a perfect square trinomial. A perfect square trinomial is an expression that can be written as $A+B$ ^ $2$ or $A-B$ ^ $2$ , where $A$ and $B$ are expressions. The standard form of a perfect square trinomial is $A^2 \pm 2AB + B^2$ .
Comparison with Standard Form: Let's compare the given expression with the standard form of a perfect square trinomial. We have $A = (x-3)$ and we need to find $B$ such that the expression matches $A^2 - 2AB + B^2$ . We can see that $A^2$ is $(x-3)^2$ , and $B^2$ should be $25$ , which means $B$ is $5$ since $5^2 = 25$ .
Matching the Middle Term: Now, we need to check if the middle term $-10(x-3)$ matches the $-2AB$ term in the perfect square trinomial pattern. For our expression, $A$ is $(x-3)$ and $B$ is $5$ , so $-2AB$ would be $-2\cdot(x-3)\cdot5$ , which simplifies to $-10(x-3)$ .
Factoring Using the Pattern: Since the given expression matches the pattern $A^2 - 2AB + B^2$ , where $A$ is $(x-3)$ and $B$ is $5$ , we can factor it using the pattern $(A-B)^2$ . Therefore, the factored form of the expression is $((x-3)-5)^2$ or $(x-3-5)^2$ , which simplifies to $(x-8)^2$ .
Successful Factoring: We have successfully factored the expression using the pattern $(A-B)^2$ . This corresponds to choice $A$ from the given options, which are $(U+V)^2$ or $(U-V)^2$ .

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