Question

match each pair of polynomials with their sum:

pairs of polynomials:

1. \\(12x^2 + 3x + 6\\) and \\(-7x^2 - 4x - 2\\)
2. \\(2x^2 - x\\) and \\(-x - 2x^2 - 2\\)
3. \\(x + x^2 + 2\\) and \\(x^2 - 2 - x\\)
4. \\(x^2 + x\\) and \\(x^2 + 8x - 2\\)

sums:

• \\(-2x - 2\\)
• \\(2x^2\\)
• \\(2x^2 + 9x - 2\\)
• \\(5x^2 - x + 4\\)

Explanation:

⚡ Using what you learned: adding and subtracting polynomials

Step 1: Simplify the first pair

Add the polynomials:

$$ (12x^2 + 3x + 6) + (-7x^2 - 4x - 2) $$

Combine like terms:

$$ (12x^2 - 7x^2) + (3x - 4x) + (6 - 2) = 5x^2 - x + 4 $$

Step 2: Simplify the second pair

Add the polynomials:

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

Combine like terms:

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

Note: Looking at the options, the closest match is \( -2x - 2 \). Let's check if the option \( 2x - 2 \) has a typo in the question's image, or if subtraction was intended. If we subtract them:

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

If we do the other way around:

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

Since \( -2x - 2 \) is the direct sum, the option \( 2x - 2 \) in the image is likely a typo for \( -2x - 2 \).

Step 3: Simplify the third pair

Add the polynomials:

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

Combine like terms:

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

Step 4: Simplify the fourth pair

Add the polynomials:

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

Combine like terms:

$$ (x^2 + x^2) + (x + 8x) - 2 = 2x^2 + 9x - 2 $$

Answer:

The correct matches are:

• \( 12x^2 + 3x + 6 \) and \( -7x^2 - 4x - 2 \) \(\longleftrightarrow\) \( 5x^2 - x + 4 \)
• \( 2x^2 - x \) and \( -x - 2x^2 - 2 \) \(\longleftrightarrow\) \( 2x - 2 \) (contains a typo in the original option, represents \( -2x - 2 \))
• \( x + x^2 + 2 \) and \( x^2 - 2 - x \) \(\longleftrightarrow\) \( 2x^2 \)
• \( x^2 + x \) and \( x^2 + 8x - 2 \) \(\longleftrightarrow\) \( 2x^2 + 9x - 2 \)