Question

question 1
type the correct answer in each box.
consider these quadratic expressions:
a. \\(-3x^2 + 11x - 3\\)
b. \\(11x^2 - x + 10\\)
c. \\(4x^2 + 27x - 28\\)
d. \\(-3x^2 + 11x + 31\\)

for each polynomial operation, write the letter that corresponds to the resulting expression from the list.
\\((5x^2 + 2x + 1) + (6x^2 - 3x + 9)\\) is equivalent to expression .
\\((-3x^2 + 6x - 12) + (5x + 9)\\) is equivalent to expression .
\\((8x + 16) - (3x^2 - 3x - 15)\\) is equivalent to expression .
\\((5x^2 + 23x - 7) - (x^2 - 4x + 21)\\) is equivalent to expression .

Explanation:

🆕 New Concept Discovered: Adding and Subtracting Polynomials
Combining like terms by grouping matching variable powers.

Step 1: Simplify the first operation

Evaluate the first expression by grouping and combining like terms:

$$ (5x^2 + 2x + 1) + (6x^2 - 3x + 9) $$

Group the \(x^2\) terms, the \(x\) terms, and the constant terms:

$$ (5x^2 + 6x^2) + (2x - 3x) + (1 + 9) $$

$$ 11x^2 - x + 10 $$

This matches expression B.

Step 2: Simplify the second operation

Evaluate the second expression:

$$ (-3x^2 + 6x - 12) + (5x + 9) $$

Group the like terms:

$$ -3x^2 + (6x + 5x) + (-12 + 9) $$

$$ -3x^2 + 11x - 3 $$

This matches expression A.

Step 3: Simplify the third operation

Evaluate the third expression, making sure to distribute the negative sign to all terms inside the second parentheses:

$$ (8x + 16) - (3x^2 - 3x - 15) $$

$$ 8x + 16 - 3x^2 + 3x + 15 $$

Group the like terms:

$$ -3x^2 + (8x + 3x) + (16 + 15) $$

$$ -3x^2 + 11x + 31 $$

This matches expression D.

Step 4: Simplify the fourth operation

Evaluate the fourth expression, distributing the negative sign:

$$ (5x^2 + 23x - 7) - (x^2 - 4x + 21) $$

$$ 5x^2 + 23x - 7 - x^2 + 4x - 21 $$

Group the like terms:

$$ (5x^2 - x^2) + (23x + 4x) + (-7 - 21) $$

$$ 4x^2 + 27x - 28 $$

This matches expression C.

Answer:

• \((5x^2 + 2x + 1) + (6x^2 - 3x + 9)\) is equivalent to expression B.
• \((-3x^2 + 6x - 12) + (5x + 9)\) is equivalent to expression A.
• \((8x + 16) - (3x^2 - 3x - 15)\) is equivalent to expression D.
• \((5x^2 + 23x - 7) - (x^2 - 4x + 21)\) is equivalent to expression C.