poopey
Answered

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



Answer :

Sure, let's solve this problem step by step:

We need to calculate the difference between the two polynomials:

[tex]\[ P_1 = x^4 + x^3 + x^2 + x \][/tex]

and

[tex]\[ P_2 = x^4 - x^3 + x^2 - x \][/tex]

To find the difference \( P_1 - P_2 \), we will subtract \( P_2 \) from \( P_1 \):

1. Write down both polynomials:
[tex]\[ P_1 = x^4 + x^3 + x^2 + x \][/tex]
[tex]\[ P_2 = x^4 - x^3 + x^2 - x \][/tex]

2. Subtract \( P_2 \) from \( P_1 \):
[tex]\[ P_1 - P_2 = (x^4 + x^3 + x^2 + x) - (x^4 - x^3 + x^2 - x) \][/tex]

3. Distribute the subtraction (subtract each term of \( P_2 \) from the corresponding term in \( P_1 \)):
[tex]\[ P_1 - P_2 = x^4 + x^3 + x^2 + x - x^4 + x^3 - x^2 + x \][/tex]

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

5. Simplify the expression to get the final result:
[tex]\[ = 2x^3 + 2x \][/tex]

So, the difference of the polynomials \( \left(x^4 + x^3 + x^2 + x\right) - \left(x^4 - x^3 + x^2 - x\right) \) is:
[tex]\[ 2x^3 + 2x \][/tex]

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