Question

the sum of two polynomials is $-yz^2 - 3z^2 - 4y + 4$. if one of the polynomials is $y - 4yz^2 - 3$, what is the other polynomial?
\\(\circ -2yz^2 - 4y + 7\\)
\\(\circ -2yz^2 - 3y + 1\\)
\\(\circ -5yz^2 + 3z^2 - 3y + 1\\)
\\(\circ 3yz^2 - 3z^2 - 5y + 7\\)

Explanation:

Step1: Define the unknown polynomial

Let the unknown polynomial be $P$.

Step2: Set up the sum equation

The sum of the two polynomials is given, so:
$$P + (y - 4yz^2 - 3) = -yz^2 - 3z^2 - 4y + 4$$

Step3: Isolate the unknown polynomial

Rearrange to solve for $P$ by subtracting the known polynomial:
$$P = (-yz^2 - 3z^2 - 4y + 4) - (y - 4yz^2 - 3)$$

Step4: Distribute the negative sign

$$P = -yz^2 - 3z^2 - 4y + 4 - y + 4yz^2 + 3$$

Step5: Combine like terms

Combine $yz^2$ terms: $-yz^2 + 4yz^2 = 3yz^2$
Combine $z^2$ terms: $-3z^2$ (no other $z^2$ terms)
Combine $y$ terms: $-4y - y = -5y$
Combine constant terms: $4 + 3 = 7$
$$P = 3yz^2 - 3z^2 - 5y + 7$$

Answer:

3yz² - 3z² - 5y + 7 (corresponding to the last option)