Which polynomial represents the difference below?

[tex]\((7x^2 + 8) - (4x^2 + x + 6)\)[/tex]

A. [tex]\(3x^2 + x + 14\)[/tex]
B. [tex]\(11x^2 + x + 14\)[/tex]
C. [tex]\(11x^2 - x + 2\)[/tex]
D. [tex]\(3x^2 - x + 2\)[/tex]



Answer :

To determine which polynomial represents the difference [tex]\((7x^2 + 8) - (4x^2 + x + 6)\)[/tex], we need to follow these steps:

1. Rewrite the expression for clarity:
[tex]\[ (7x^2 + 8) - (4x^2 + x + 6) \][/tex]

2. Distribute the negative sign through the second polynomial:
[tex]\[ 7x^2 + 8 - 4x^2 - x - 6 \][/tex]

3. Combine like terms:
Group the [tex]\(x^2\)[/tex] terms: [tex]\(7x^2 - 4x^2\)[/tex]
Group the [tex]\(x\)[/tex] terms: [tex]\(-x\)[/tex] (no other [tex]\(x\)[/tex] term to combine with)
Group the constant terms: [tex]\(8 - 6\)[/tex]

4. Simplify the grouped terms:
Combine the [tex]\(x^2\)[/tex] terms: [tex]\(7x^2 - 4x^2 = 3x^2\)[/tex]
Keep the [tex]\(x\)[/tex] term as is: [tex]\(-x\)[/tex]
Combine the constants: [tex]\(8 - 6 = 2\)[/tex]

5. Write the simplified polynomial:
[tex]\[ 3x^2 - x + 2 \][/tex]

Therefore, the polynomial representing the difference is:
[tex]\[ 3x^2 - x + 2 \][/tex]

Matching this with the given choices, the correct answer is:
[tex]\[ \boxed{D. \, 3x^2 - x + 2} \][/tex]