Question

select the correct answer.
the sum of two trinomials is (7x^2 - 5x + 4). if one of the trinomials is (3x^2 + 2x - 1), then what is the other trinomial?
a. (4x^2 - 3x + 3)
b. (10x^2 - 3x + 3)
c. (4x^2 - 7x + 5)
d. (10x^2 + 7x + 5)

Explanation:

Step1: Let the other trinomial be \( y \). We know that the sum of the two trinomials is \( 7x^2 - 5x + 4 \) and one trinomial is \( 3x^2 + 2x - 1 \). So, we can write the equation: \( (3x^2 + 2x - 1) + y = 7x^2 - 5x + 4 \).

Step2: To find \( y \), we solve for it by subtracting \( 3x^2 + 2x - 1 \) from both sides: \( y = (7x^2 - 5x + 4) - (3x^2 + 2x - 1) \).

Step3: Distribute the negative sign: \( y = 7x^2 - 5x + 4 - 3x^2 - 2x + 1 \).

Step4: Combine like terms. For the \( x^2 \) terms: \( 7x^2 - 3x^2 = 4x^2 \). For the \( x \) terms: \( -5x - 2x = -7x \). For the constant terms: \( 4 + 1 = 5 \). So, \( y = 4x^2 - 7x + 5 \)? Wait, no, wait. Wait, let's check again. Wait, the sum is \( 7x^2 -5x +4 \), and one term is \( 3x^2 +2x -1 \). So subtracting: \( (7x^2 -5x +4) - (3x^2 +2x -1) = 7x^2 -5x +4 -3x^2 -2x +1 = (7x^2 -3x^2) + (-5x -2x) + (4 +1) = 4x^2 -7x +5 \)? But that's option C? Wait, no, wait, maybe I made a mistake. Wait, no, let's check the options again. Wait, option A is \( 4x^2 -3x +3 \), option C is \( 4x^2 -7x +5 \). Wait, let's recalculate:

\( 7x^2 -5x +4 - (3x^2 +2x -1) = 7x^2 -5x +4 -3x^2 -2x +1 = (7-3)x^2 + (-5-2)x + (4+1) = 4x^2 -7x +5 \). So that's option C? Wait, but let's check by adding option C to \( 3x^2 +2x -1 \): \( (3x^2 +2x -1) + (4x^2 -7x +5) = 3x^2 +4x^2 +2x -7x -1 +5 = 7x^2 -5x +4 \), which matches the sum. So the correct answer is option C. Wait, but earlier I thought maybe I messed up, but the calculation checks out.

Wait, but let's check the options again. The options are:

A. \( 4x^2 -3x +3 \)

B. \( 10x^2 -3x +3 \)

C. \( 4x^2 -7x +5 \)

D. \( 10x^2 +7x +5 \)

So when we add \( 3x^2 +2x -1 \) and \( 4x^2 -7x +5 \), we get \( 7x^2 -5x +4 \), which is the given sum. So the correct answer is C.

Wait, but let's do the subtraction again:

Sum: \( 7x^2 -5x +4 \)

One term: \( 3x^2 +2x -1 \)

Other term: Sum - One term = \( (7x^2 -5x +4) - (3x^2 +2x -1) \)

= \( 7x^2 -5x +4 -3x^2 -2x +1 \)

= \( (7x^2 -3x^2) + (-5x -2x) + (4 +1) \)

= \( 4x^2 -7x +5 \)

Yes, that's correct. So the other trinomial is \( 4x^2 -7x +5 \), which is option C.

Answer:

C. \( 4x^2 - 7x + 5 \)