Question

select the correct answer.

the difference of two trinomials is (x^2 - 10x + 2). if one of the trinomials is (3x^2 - 11x - 4), then which expression could represent the other trinomial?

(2x^2 - x - 2)
(2x^2 + x + 6)
(4x^2 + 21x + 6)
(4x^2 - 21x - 2)

Explanation:

⚡ Using what you learned: adding and subtracting polynomials

Step 1: Set up the algebraic relationship

Let the unknown trinomial be \( T \). The difference between the two trinomials is given as \( x^2 - 10x + 2 \). This gives two possible scenarios:

1. \( (3x^2 - 11x - 4) - T = x^2 - 10x + 2 \)
2. \( T - (3x^2 - 11x - 4) = x^2 - 10x + 2 \)

We will test the first scenario to see if it matches any of the given options.

Step 2: Solve for the unknown trinomial in Case 1

$$ T = (3x^2 - 11x - 4) - (x^2 - 10x + 2) $$

Distribute the negative sign:

$$ T = 3x^2 - 11x - 4 - x^2 + 10x - 2 $$

Combine like terms:

$$ T = (3x^2 - x^2) + (-11x + 10x) + (-4 - 2) $$

$$ T = 2x^2 - x - 6 $$

This trinomial is not among the choices.

Step 3: Solve for the unknown trinomial in Case 2

$$ T = (x^2 - 10x + 2) + (3x^2 - 11x - 4) $$

Combine like terms:

$$ T = (x^2 + 3x^2) + (-10x - 11x) + (2 - 4) $$

$$ T = 4x^2 - 21x - 2 $$

This matches the fourth option.

Answer:

\( 4x^2 - 21x - 2 \)