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What is the difference of the polynomials?

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

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



Answer :

GinnyAnswer
To find the difference between two given polynomials [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex], follow these steps:

First, identify the polynomials:
[tex]\[ P(x) = x^4 + x^3 + x^2 + x \][/tex]
[tex]\[ Q(x) = x^4 - x^3 + x^2 - x \][/tex]

Next, subtract [tex]\( Q(x) \)[/tex] from [tex]\( P(x) \)[/tex]:
[tex]\[ P(x) - Q(x) = (x^4 + x^3 + x^2 + x) - (x^4 - x^3 + x^2 - x) \][/tex]

Distribute the negative sign across the terms of [tex]\( Q(x) \)[/tex]:
[tex]\[ P(x) - Q(x) = x^4 + x^3 + x^2 + x - x^4 + x^3 - x^2 + x \][/tex]

Combine like terms:
[tex]\[ P(x) - Q(x) = (x^4 - x^4) + (x^3 + x^3) + (x^2 - x^2) + (x + x) \][/tex]

Simplify each set of like terms:
[tex]\[ P(x) - Q(x) = 0 + 2x^3 + 0 + 2x \][/tex]

So, the difference of the polynomials is:
[tex]\[ 2x^3 + 2x \][/tex]

Therefore, the correct answer is:
[tex]\[ 2x^3 + 2x \][/tex]

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