The difference between two trinomials is [tex]5x^2 + 11x - 16[/tex]. If one of the trinomials is [tex]3x^2 - 2x + 7[/tex], then which expression could be the other trinomial?

A. [tex]2x^2 + 13x - 23[/tex]

B. [tex]8x^2 + 9x - 9[/tex]

C. [tex]8x^2 + 13x - 23[/tex]

D. [tex]2x^2 + 9x - 9[/tex]



Answer :

To find the other trinomial, we need to determine a trinomial that, when subtracted from the given trinomial [tex]\(3x^2 - 2x + 7\)[/tex], results in the difference [tex]\(5x^2 + 11x - 16\)[/tex].

Let's denote the other trinomial as [tex]\(Ax^2 + Bx + C\)[/tex]. According to the problem, the equation we are dealing with is:

[tex]\[ (Ax^2 + Bx + C) - (3x^2 - 2x + 7) = 5x^2 + 11x - 16 \][/tex]

We can rewrite this equation to separate the coefficients:

1. For the [tex]\(x^2\)[/tex] term:
[tex]\[ A - 3 = 5 \][/tex]
2. For the [tex]\(x\)[/tex] term:
[tex]\[ B - (-2) = 11 \implies B + 2 = 11 \][/tex]
3. For the constant term:
[tex]\[ C - 7 = -16 \][/tex]

Now we solve these equations one by one:

1. For the [tex]\(x^2\)[/tex] term:
[tex]\[ A - 3 = 5 \implies A = 5 + 3 = 8 \][/tex]

2. For the [tex]\(x\)[/tex] term:
[tex]\[ B + 2 = 11 \implies B = 11 - 2 = 9 \][/tex]

3. For the constant term:
[tex]\[ C - 7 = -16 \implies C = -16 + 7 = -9 \][/tex]

Thus, the other trinomial is:
[tex]\[ 8x^2 + 9x - 9 \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{8x^2 + 9x - 9} \][/tex]

Option B is correct.