High School

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

A. [tex]7x^3 - 5x^2 - x - 7[/tex]
B. [tex]7x^3 - 5x^2 + x - 7[/tex]
C. [tex]7x^3 + x^2 - 5x - 7[/tex]
D. [tex]7x^8 - 4x^2 - 7[/tex]

Answer :

Sure! Let's simplify the polynomial expression step-by-step:

We need to simplify [tex]\((4x^3 + 6x - 7) + (3x^3 - 5x^2 - 5x)\)[/tex].

1. Identify and Group Like Terms:

- Cubic terms ([tex]\(x^3\)[/tex]):
- From the first polynomial: [tex]\(4x^3\)[/tex]
- From the second polynomial: [tex]\(3x^3\)[/tex]

Combine them:
[tex]\[
4x^3 + 3x^3 = 7x^3
\][/tex]

2. Quadratic terms ([tex]\(x^2\)[/tex]):

- From the first polynomial: There is no [tex]\(x^2\)[/tex] term (so it is [tex]\(0x^2\)[/tex])
- From the second polynomial: [tex]\(-5x^2\)[/tex]

Combine them:
[tex]\[
0x^2 - 5x^2 = -5x^2
\][/tex]

3. Linear terms ([tex]\(x\)[/tex]):

- From the first polynomial: [tex]\(6x\)[/tex]
- From the second polynomial: [tex]\(-5x\)[/tex]

Combine them:
[tex]\[
6x - 5x = 1x \quad \text{which can be written as} \quad x
\][/tex]

4. Constant terms:

- From the first polynomial: [tex]\(-7\)[/tex]
- From the second polynomial: There is no constant term (0)

Combine them:
[tex]\[
-7 + 0 = -7
\][/tex]

Now, let's combine all the simplified terms together:
[tex]\[
7x^3 - 5x^2 + x - 7
\][/tex]

Therefore, the simplest form of the given expression is:
[tex]\[
7x^3 - 5x^2 + x - 7
\][/tex]

This matches with option B: [tex]\(7x^3 - 5x^2 + x - 7\)[/tex].

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