The sum of two polynomials is 10a2b2 – 8a2b 6ab2 – 4ab 2. if one addend is –5a2b2 12a2b – 5, what is the other addend?
The Correct answer and Explanation is:
To find the other addend of the polynomial sum ( P(x) = 10a^2b^2 – 8a^2b + 6ab^2 – 4ab^2 ), given one of the addends ( Q(x) = -5a^2b^2 + 12a^2b – 5 ), we can set up the equation as follows:
[
P(x) = Q(x) + R(x)
]
where ( R(x) ) is the unknown polynomial we need to determine. Rearranging the equation gives us:
[
R(x) = P(x) – Q(x)
]
Substituting the known polynomials into this equation, we have:
[
R(x) = (10a^2b^2 – 8a^2b + 6ab^2 – 4ab^2) – (-5a^2b^2 + 12a^2b – 5)
]
Next, we simplify the expression step-by-step. First, distribute the negative sign through ( Q(x) ):
[
R(x) = 10a^2b^2 – 8a^2b + 6ab^2 – 4ab^2 + 5a^2b^2 – 12a^2b + 5
]
Now, we can combine like terms:
- Combine the ( a^2b^2 ) terms:
[
10a^2b^2 + 5a^2b^2 = 15a^2b^2
] - Combine the ( a^2b ) terms:
[
-8a^2b – 12a^2b = -20a^2b
] - Combine the ( ab^2 ) terms:
[
6ab^2 – 4ab^2 = 2ab^2
] - The constant term:
[
5
]
Putting it all together, we find:
[
R(x) = 15a^2b^2 – 20a^2b + 2ab^2 + 5
]
Thus, the other addend ( R(x) ) is:
[
\boxed{15a^2b^2 – 20a^2b + 2ab^2 + 5}
]
Explanation
In polynomial addition, we combine the coefficients of like terms. In this case, we started with the total sum and subtracted the known addend to isolate the unknown addend. The process involved carefully distributing the negative sign and then combining terms systematically to ensure accuracy.
By organizing the like terms, we identified terms based on their degrees and variables, which allowed us to simplify effectively. This approach not only yielded the other addend but also reinforced our understanding of polynomial manipulation.
Overall, this method of rearranging equations and systematically simplifying terms is foundational in algebra, particularly when working with polynomials. It emphasizes the importance of maintaining the correct signs and accurately combining coefficients, which are crucial skills for solving polynomial equations.