The sum of two polynomials is $10 a^2 b^2-8 a^2 b+6 a b^2-4 a b+2$. If one addend is $-5 a^2 b^2+12 a^2 b-5$, what is the other addend?
Answer (2)
To find the other addend of a polynomial sum, we calculate P 2 as P 2 = S − P 1 . After combining like terms, we find that the other addend is 15 a 2 b 2 − 20 a 2 b + 6 a b 2 − 4 ab + 7 .
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We have the sum of two polynomials S = 10 a 2 b 2 − 8 a 2 b + 6 a b 2 − 4 ab + 2 and one of the polynomials P 1 = − 5 a 2 b 2 + 12 a 2 b − 5 .
We need to find the other polynomial P 2 , where P 2 = S − P 1 .
Subtracting P 1 from S , we have P 2 = ( 10 a 2 b 2 − 8 a 2 b + 6 a b 2 − 4 ab + 2 ) − ( − 5 a 2 b 2 + 12 a 2 b − 5 ) .
Combining like terms, we get P 2 = 15 a 2 b 2 − 20 a 2 b + 6 a b 2 − 4 ab + 7 . Therefore, the other addend is 15 a 2 b 2 − 20 a 2 b + 6 a b 2 − 4 ab + 7 .
Explanation
Problem Analysis We are given that the sum of two polynomials is 10 a 2 b 2 − 8 a 2 b + 6 a b 2 − 4 ab + 2 . One of the polynomials is − 5 a 2 b 2 + 12 a 2 b − 5 . We need to find the other polynomial.
Setting up the equation Let the sum of the polynomials be S and the given polynomial be P 1 . Let the polynomial we need to find be P 2 . Then, we have S = P 1 + P 2 We are given S = 10 a 2 b 2 − 8 a 2 b + 6 a b 2 − 4 ab + 2 P 1 = − 5 a 2 b 2 + 12 a 2 b − 5 We need to find P 2 .
Isolating the unknown polynomial From the equation S = P 1 + P 2 , we can write P 2 = S − P 1 Substituting the given values, we get P 2 = ( 10 a 2 b 2 − 8 a 2 b + 6 a b 2 − 4 ab + 2 ) − ( − 5 a 2 b 2 + 12 a 2 b − 5 ) P 2 = 10 a 2 b 2 − 8 a 2 b + 6 a b 2 − 4 ab + 2 + 5 a 2 b 2 − 12 a 2 b + 5 Now, we combine the like terms.
Combining like terms Combining the a 2 b 2 terms: 10 a 2 b 2 + 5 a 2 b 2 = 15 a 2 b 2 Combining the a 2 b terms: − 8 a 2 b − 12 a 2 b = − 20 a 2 b The a b 2 term is just 6 a b 2 .
The ab term is just − 4 ab .
Combining the constant terms: 2 + 5 = 7 Therefore, P 2 = 15 a 2 b 2 − 20 a 2 b + 6 a b 2 − 4 ab + 7
Final Answer The other addend is 15 a 2 b 2 − 20 a 2 b + 6 a b 2 − 4 ab + 7 .
Examples
Polynomials are used in various fields such as engineering, physics, economics, and computer science. For example, engineers use polynomials to model curves and surfaces, physicists use them to describe the motion of objects, economists use them to model economic growth, and computer scientists use them to design algorithms. In real life, if you were designing a bridge, you might use polynomials to model the curve of the arch. Or, if you were an economist, you might use polynomials to predict how the price of a product will change over time.
