Find: $(4 x^2 y^3+2 x y^2-2 y)-(-7 x^2 y^3+6 x y^2-2 y)$. Place the correct coefficients in the difference.
Answer (2)
To find the difference between the two polynomials, we first distribute the negative sign to the second polynomial, resulting in 7 x 2 y 3 − 6 x y 2 + 2 y . Then, by combining like terms, we obtain 11 x 2 y 3 − 4 x y 2 with coefficients 11 and -4, respectively.
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Distribute the negative sign: − ( − 7 x 2 y 3 + 6 x y 2 − 2 y ) = 7 x 2 y 3 − 6 x y 2 + 2 y .
Combine like terms: ( 4 x 2 y 3 + 2 x y 2 − 2 y ) + ( 7 x 2 y 3 − 6 x y 2 + 2 y ) .
Simplify: 11 x 2 y 3 − 4 x y 2 .
The coefficients are 11 for x 2 y 3 and − 4 for x y 2 .
Explanation
Understanding the Problem We are asked to subtract the polynomial ( − 7 x 2 y 3 + 6 x y 2 − 2 y ) from the polynomial ( 4 x 2 y 3 + 2 x y 2 − 2 y ) . This involves distributing the negative sign to each term of the second polynomial and then combining like terms.
Distributing the Negative Sign First, distribute the negative sign to each term in the second polynomial:
− ( − 7 x 2 y 3 + 6 x y 2 − 2 y ) = 7 x 2 y 3 − 6 x y 2 + 2 y
Combining Like Terms Now, combine the like terms from both polynomials:
( 4 x 2 y 3 + 2 x y 2 − 2 y ) + ( 7 x 2 y 3 − 6 x y 2 + 2 y )
Adding x 2 y 3 Terms Add the coefficients of the x 2 y 3 terms:
4 x 2 y 3 + 7 x 2 y 3 = ( 4 + 7 ) x 2 y 3 = 11 x 2 y 3
Adding x y 2 Terms Add the coefficients of the x y 2 terms:
2 x y 2 − 6 x y 2 = ( 2 − 6 ) x y 2 = − 4 x y 2
Adding y Terms Add the coefficients of the y terms:
− 2 y + 2 y = ( − 2 + 2 ) y = 0 y = 0
Final Result Write the resulting polynomial:
11 x 2 y 3 − 4 x y 2 + 0 = 11 x 2 y 3 − 4 x y 2
The coefficients are 11 for x 2 y 3 and -4 for x y 2 .
Examples
Polynomial subtraction is used in various fields such as engineering, physics, and computer graphics. For example, in computer graphics, subtracting polynomials can help determine the difference between two curves or surfaces, which is useful for rendering and modeling objects. In physics, it can be used to find the net force acting on an object when multiple forces are involved. Understanding polynomial subtraction allows for efficient problem-solving in these areas.
