The sum of two polynomials is $-y z^2-3 z^2-4 y+4$. If one of the polynomials is $y-4 y z^2-3$, what is the other polynomial?
A. $-2 y z^2-4 y+7$
B. $-2 y z^2-3 y+1$
C. $-5 y z^2+3 z^2-3 y+1$
D. $3 y z^2-3 z^2-5 y+7$
Answer (2)
To find the other polynomial, we calculated that P 2 = 3 y z 2 − 3 z 2 − 5 y + 7 . This matches with option D from the choices provided. Therefore, the answer is option D.
;
We have two polynomials P 1 and P 2 such that P 1 + P 2 = − y z 2 − 3 z 2 − 4 y + 4 .
Given P 1 = y − 4 y z 2 − 3 , we want to find P 2 .
We have P 2 = ( − y z 2 − 3 z 2 − 4 y + 4 ) − ( y − 4 y z 2 − 3 ) .
Simplifying the expression, we get P 2 = 3 y z 2 − 3 z 2 − 5 y + 7 .
3 y z 2 − 3 z 2 − 5 y + 7
Explanation
Problem Analysis We are given that the sum of two polynomials is − y z 2 − 3 z 2 − 4 y + 4 . One of the polynomials is y − 4 y z 2 − 3 . We need to find the other polynomial. Let the first polynomial be P 1 = y − 4 y z 2 − 3 and the second polynomial be P 2 . We are given that P 1 + P 2 = − y z 2 − 3 z 2 − 4 y + 4 . Therefore, P 2 = ( − y z 2 − 3 z 2 − 4 y + 4 ) − ( y − 4 y z 2 − 3 ) .
Simplifying the Expression Now, we simplify the expression for P 2 by combining like terms:
P 2 = − y z 2 − 3 z 2 − 4 y + 4 − y + 4 y z 2 + 3
P 2 = ( − y z 2 + 4 y z 2 ) − 3 z 2 + ( − 4 y − y ) + ( 4 + 3 )
P 2 = 3 y z 2 − 3 z 2 − 5 y + 7
Finding the Other Polynomial Therefore, the other polynomial is 3 y z 2 − 3 z 2 − 5 y + 7 .
Examples
Polynomials are used in various fields such as physics, engineering, and computer science. For example, in physics, polynomials can be used to model the trajectory of a projectile. In engineering, they can be used to design curves and surfaces. In computer science, they are used in cryptography and data compression. Understanding how to manipulate polynomials, such as adding and subtracting them, is essential for solving problems in these fields.
