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What is the difference of the polynomials?

[tex]\left(x^4+x^3+x^2+x\right) - \left(x^4-x^3+x^3-x\right)[/tex]

A. [tex]2x^2[/tex]
B. [tex]2x^3 + 2[/tex]
C. [tex]x^6 + x^2[/tex]
D. [tex]2x^6 + 2x^2[/tex]

Answer :

To find the difference between the polynomials [tex]\((x^4 + x^3 + x^2 + x)\)[/tex] and [tex]\((x^4 - x^3 + x^3 - x)\)[/tex], we need to subtract the second polynomial from the first.

1. Write down the polynomials:

- First polynomial: [tex]\(x^4 + x^3 + x^2 + x\)[/tex]
- Second polynomial: [tex]\(x^4 - x^3 + x^3 - x\)[/tex]

2. Subtract the second polynomial from the first:

- Align like terms:
[tex]\[
(x^4 + x^3 + x^2 + x) - (x^4 - x^3 + x^3 - x)
\][/tex]

- Distribute the negative sign to the second polynomial:
[tex]\[
= x^4 + x^3 + x^2 + x - x^4 + x^3 - x^3 + x
\][/tex]

3. Simplify the expression:

- Combine the like terms:
- [tex]\(x^4 - x^4 = 0\)[/tex]
- [tex]\(x^3 + x^3 - x^3 = x^3\)[/tex]
- [tex]\(x^2\)[/tex] (only in the first polynomial)
- [tex]\(x + x = 2x\)[/tex]

- Thus, the simplified expression is:
[tex]\[
x^3 + x^2 + 2x
\][/tex]

4. Factor the expression if possible:

- The expression [tex]\(x^3 + x^2 + 2x\)[/tex] can be factored by taking out the greatest common factor, [tex]\(x\)[/tex]:
[tex]\[
= x(x^2 + x + 2)
\][/tex]

So, the difference of the polynomials is [tex]\(x(x^2 + x + 2)\)[/tex].

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