Question

lucia finds that \\((3x^2 + 3x + 5) + (7x^2 - 9x + 8) = 10x^2 - 12x + 13\\). what error did lucia make?

• she found the difference instead of the sum.
• she combined the terms \\(3x^2\\) and \\(7x^2\\) incorrectly.
• she combined the terms \\(3x\\) and \\(-9x\\) incorrectly.
• she did not combine all like terms.

Explanation:

⚡ Using what you learned: adding and subtracting polynomials

Step 1: Group like terms

Group the terms of the expression by their variable powers:

$$ (3x^2 + 7x^2) + (3x - 9x) + (5 + 8) $$

Step 2: Simplify each group

Calculate the sum for each group of like terms:

• Quadratic terms: \( 3x^2 + 7x^2 = 10x^2 \)
• Linear terms: \( 3x + (-9x) = -6x \)
• Constant terms: \( 5 + 8 = 13 \)

The correct simplified expression is:

$$ 10x^2 - 6x + 13 $$

Step 3: Identify the error

Lucia's result is \( 10x^2 - 12x + 13 \).

Comparing the linear terms:

• Correct: \( 3x + (-9x) = -6x \)
• Lucia's result: \( -12x \)

Lucia calculated \( -3x - 9x = -12x \) instead of adding \( 3x \) and \( -9x \). Therefore, she combined the terms \( 3x \) and \( -9x \) incorrectly.

Answer:

She combined the terms \( 3x \) and \( -9x \) incorrectly.