Question

a student factors \\(10x^2 + 3x - 27\\) to the following.

\\((2x - 3)(5x + 9)\\)

which statement about \\((2x - 3)(5x + 9)\\) is true?

• the expression is equivalent, and it is completely factored.
• the expression is equivalent, but it is not completely factored.
• the expression is not equivalent, but it is completely factored.
• the expression is not equivalent, and it is not completely factored.

Explanation:

Expand the factored expression

$$ (2x - 3)(5x + 9) = 2x(5x) + 2x(9) - 3(5x) - 3(9) $$

$$ = 10x^2 + 18x - 15x - 27 = 10x^2 + 3x - 27 $$

Check equivalence and completeness

$$ 10x^2 + 3x - 27 \equiv 10x^2 + 3x - 27 $$

The binomials \(2x - 3\) and \(5x + 9\) have integer coefficients with no common factors other than 1, meaning they cannot be factored further.

Answer:

• The expression is equivalent, and it is completely factored. (Correct answer)
• The expression is equivalent, but it is not completely factored.
• The expression is not equivalent, but it is completely factored.
• The expression is not equivalent, and it is not completely factored.