What is the difference of the polynomials? (x ^ 4 + x ^ 3 + x ^ 2 + x) – (x ^ 4 – x ^ 3 + x ^ 2 – x)
A 2×2
B 2x ^ 3 + 2x
C x ^ 6 + x ^ 2
D 2x ^ 6 + 2x ^ 2
The Correct Answer and Explanation is :
The given expression is:
[
(x^4 + x^3 + x^2 + x) – (x^4 – x^3 + x^2 – x)
]
To find the difference between these two polynomials, you subtract the second polynomial from the first. To simplify this, distribute the minus sign to each term inside the parentheses of the second polynomial:
[
x^4 + x^3 + x^2 + x – x^4 + x^3 – x^2 + x
]
Now, group like terms:
[
(x^4 – x^4) + (x^3 + x^3) + (x^2 – x^2) + (x + x)
]
Simplifying each group:
- ( x^4 – x^4 = 0 )
- ( x^3 + x^3 = 2x^3 )
- ( x^2 – x^2 = 0 )
- ( x + x = 2x )
The simplified result is:
[
2x^3 + 2x
]
Thus, the correct answer is:
B: ( 2x^3 + 2x )
Explanation:
When subtracting polynomials, you need to carefully distribute the negative sign across all terms of the second polynomial. This changes the signs of all the terms in the second polynomial, allowing you to combine like terms from both polynomials.
In this problem, the highest degree term ( x^4 ) cancels out since both the first and second polynomials contain ( x^4 ) but with opposite signs. The terms ( x^3 ) and ( x ) double, and the terms ( x^2 ) also cancel out. The final result is ( 2x^3 + 2x ), which reflects the combination of remaining non-zero like terms.