What is the difference of the polynomials? (x^4 + x^3 + x^2 + x) - (x^4 - x^3 + x^2...
Check the final answer first, then review the worked steps.
Problem
What is the difference of the polynomials? (x^4 + x^3 + x^2 + x) - (x^4 - x^3 + x^2 - x)
Answer
\(2x^3 + 2x\)
Step-by-step solution
1. Distribute the negative sign: The problem asks for the difference between two polynomials. This means we need to subtract the second polynomial from the first. When subtracting a polynomial, we distribute the negative sign to each term in the second polynomial. This changes the sign of each term in the second polynomial.
$$(x^4 + x^3 + x^2 + x) - (x^4 - x^3 + x^2 - x) = x^4 + x^3 + x^2 + x - x^4 + x^3 - x^2 + x$$
2. Group like terms: Now that the negative sign has been distributed, we can group the terms that have the same variable and exponent. This makes it easier to combine them.
$$(x^4 - x^4) + (x^3 + x^3) + (x^2 - x^2) + (x + x)$$
3. Combine like terms: Finally, we combine the coefficients of the like terms.
- For the $x^4$ terms: $x^4 - x^4 = 0$
- For the $x^3$ terms: $x^3 + x^3 = 2x^3$
- For the $x^2$ terms: $x^2 - x^2 = 0$
- For the $x$ terms: $x + x = 2x
Combining these results, we get: $0 + 2x^3 + 0 + 2x = 2x^3 + 2x$.