The sum of two polynomials is 10a2b2 – 8a2b + 6ab2 – 4ab + 2. If one addend is –5a2b2 + 12a2b – 5, what is the other addend? 15a2b2 – 20a2b + 6ab2 – 4ab + 7 5a2b2 – 20a2b2 + 7 5a2b2 + 4a2b2 + 6ab – 4ab – 3 –15a2b2 + 20a2b2 – 6ab + 4ab – 7
The Correct Answer and Explanation is :
To find the other addend of the polynomials, we start with the equation:
[
\text{Sum} = \text{Addend}_1 + \text{Addend}_2
]
Given that the sum of the two polynomials is:
[
10a^2b^2 – 8a^2b + 6ab^2 – 4ab + 2
]
And one of the addends is:
[
\text{Addend}_1 = -5a^2b^2 + 12a^2b – 5
]
We need to determine the other addend, which we can denote as ( \text{Addend}_2 ). We can rearrange the equation to find ( \text{Addend}_2 ):
[
\text{Addend}_2 = \text{Sum} – \text{Addend}_1
]
Substituting in the known values:
[
\text{Addend}_2 = (10a^2b^2 – 8a^2b + 6ab^2 – 4ab + 2) – (-5a^2b^2 + 12a^2b – 5)
]
Now, let’s simplify the expression for ( \text{Addend}_2 ):
- Distribute the negative sign in front of ( \text{Addend}_1 ):
[
\text{Addend}_2 = 10a^2b^2 – 8a^2b + 6ab^2 – 4ab + 2 + 5a^2b^2 – 12a^2b + 5
]
- Combine like terms:
- For ( a^2b^2 ):
[
10a^2b^2 + 5a^2b^2 = 15a^2b^2
] - For ( a^2b ):
[
-8a^2b – 12a^2b = -20a^2b
] - For ( ab^2 ):
[
6ab^2 \quad (\text{only term})
] - For ( ab ):
[
-4ab \quad (\text{only term})
] - For the constant:
[
2 + 5 = 7
]
Putting it all together:
[
\text{Addend}_2 = 15a^2b^2 – 20a^2b + 6ab^2 – 4ab + 7
]
The expression we found does not match the choices directly, but if we list the choices:
- ( 15a^2b^2 – 20a^2b + 6ab^2 – 4ab + 7 )
- ( 5a^2b^2 – 20a^2b + 7 )
- ( 5a^2b^2 + 4a^2b^2 + 6ab – 4ab – 3 )
- ( -15a^2b^2 + 20a^2b – 6ab + 4ab – 7 )
The correct answer is the first option:
[
15a^2b^2 – 20a^2b + 6ab^2 – 4ab + 7
]
Explanation
In this problem, we use algebraic manipulation to isolate one polynomial from a sum. The key steps involved correctly combining like terms, ensuring we correctly apply signs, especially when distributing negatives. The solution highlights the importance of careful organization when working with polynomials. Each polynomial term is treated based on its degree and variable coefficients, allowing us to systematically find the desired addend. This technique is foundational in algebra, particularly in polynomial arithmetic, where precision is vital for correct results.