High School

What is the simplest form of [tex]\left(4 x^3+6 x-7\right)+\left(3 x^3-5 x^2-5 x+9\right) ?[/tex]

A. [tex]7 x^3-5 x^2-x+2[/tex]
B. [tex]7 x^3-5 x^2+x+2[/tex]
C. [tex]7 x^3+x^2-5 x+2[/tex]
D. [tex]7 x^6-4 x^2+2[/tex]

Answer :

To simplify the expression [tex]\((4x^3 + 6x - 7) + (3x^3 - 5x^2 - 5x + 9)\)[/tex], follow these steps:

1. Combine Similar Terms:
- Cubic term ([tex]\(x^3\)[/tex]): The first polynomial has [tex]\(4x^3\)[/tex] and the second polynomial has [tex]\(3x^3\)[/tex]. Add these together:
[tex]\[4x^3 + 3x^3 = 7x^3\][/tex]

- Quadratic term ([tex]\(x^2\)[/tex]): The first polynomial has no [tex]\(x^2\)[/tex] term, which is equivalent to [tex]\(0x^2\)[/tex]. The second polynomial has [tex]\(-5x^2\)[/tex]. Add these together:
[tex]\[0x^2 - 5x^2 = -5x^2\][/tex]

- Linear term ([tex]\(x\)[/tex]): The first polynomial has [tex]\(6x\)[/tex] and the second polynomial has [tex]\(-5x\)[/tex]. Add these together:
[tex]\[6x - 5x = x\][/tex]

- Constant term: The first polynomial has [tex]\(-7\)[/tex], and the second polynomial has [tex]\(9\)[/tex]. Add these together:
[tex]\[-7 + 9 = 2\][/tex]

2. Write the Simplified Polynomial: After combining all like terms, you get:
[tex]\[7x^3 - 5x^2 + x + 2\][/tex]

3. Choose the Correct Answer: The expression simplifies to [tex]\(7x^3 - 5x^2 + x + 2\)[/tex], which corresponds to option B: [tex]\(7x^3 - 5x^2 + x + 2\)[/tex]. However, there might be an error in the options you provided, as option B in your list doesn't match exactly with what's calculated here. The closest match is option A: [tex]\(7x^3 - 5x^2 - x + 2\)[/tex], but note that the linear term should actually be [tex]\(x\)[/tex], not [tex]\(-x\)[/tex].

The correct representation of your simplified polynomial should be a match to either the options provided or it might require a re-evaluation of the given options. Please verify with the intended correct choice.