Question

simplify the expression \\(3x(x - 12x) + 3x^2 - 2(x - 2)^2\\). which statements are true about the process and simplified product? select three options.

the term \\(-2(x - 2)^2\\) is simplified by first squaring the expression \\(x - 2\\).
the simplified product is a binomial.
after multiplying, the like terms are combined by adding and subtracting.
the parentheses are eliminated through multiplication.
the final simplified product is \\(-28x^2 + 8x - 8\\).

Explanation:

Expand the first term

Using the Distributive Property knowledge point

$$ 3x(x - 12x) = 3x(-11x) = -33x^2 $$

Expand the squared binomial

Using the Perfect Square Trinomial knowledge point

$$ (x - 2)^2 = x^2 - 4x + 4 $$

Distribute the constant factor

Using the Distributive Property knowledge point

$$ -2(x - 2)^2 = -2(x^2 - 4x + 4) = -2x^2 + 8x - 8 $$

Combine all terms

Using the Like Terms knowledge point

$$
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$$

Evaluate the statements

Using the Polynomial Classification knowledge point

• Statement 1: True. Order of operations requires squaring \(x-2\) first.
• Statement 2: False. The result \(-32x^2 + 8x - 8\) is a trinomial.
• Statement 3: True. Like terms are combined by adding and subtracting.
• Statement 4: True. Parentheses are eliminated by distributing.
• Statement 5: False. The final product is \(-32x^2 + 8x - 8\).

Answer:

• The term \(-2(x-2)^2\) is simplified by first squaring the expression \(x-2\). (Correct answer)
• The simplified product is a binomial.
• After multiplying, the like terms are combined by adding and subtracting. (Correct answer)
• The parentheses are eliminated through multiplication. (Correct answer)
• The final simplified product is \(-28x^2 +8x -8\).