Question

the sum of two polynomials is $8d^5 - 3c^3d^2 + 5c^2d^3 - 4cd^4 + 9$. if one addend is $2d^5 - c^3d^2 + 8cd^4 + 1$, what is the other addend?
○ $6d^5 - 2c^3d^2 + 5c^2d^3 - 12cd^4 + 8$
○ $6d^5 - 4c^3d^2 + 3c^2d^3 - 4cd^4 + 8$
○ $6d^5 - 4c^3d^2 + 5c^2d^3 - 12cd^4 + 8$
○ $6d^5 - 2c^3d^2 - 3c^2d^3 - 4cd^4 + 8$

Explanation:

Step1: Recall the formula for subtracting polynomials (other addend = sum - one addend)

The sum is \( 8d^5 - 3c^3d^2 + 5c^2d^3 - 4cd^4 + 9 \) and one addend is \( 2d^5 - c^3d^2 + 8cd^4 + 1 \). So we need to compute \( (8d^5 - 3c^3d^2 + 5c^2d^3 - 4cd^4 + 9) - (2d^5 - c^3d^2 + 8cd^4 + 1) \).

Step2: Distribute the negative sign to the second polynomial

\( 8d^5 - 3c^3d^2 + 5c^2d^3 - 4cd^4 + 9 - 2d^5 + c^3d^2 - 8cd^4 - 1 \)

Step3: Combine like terms for \( d^5 \) terms

\( 8d^5 - 2d^5 = 6d^5 \)

Step4: Combine like terms for \( c^3d^2 \) terms

\( -3c^3d^2 + c^3d^2 = -2c^3d^2 \)

Step5: Combine like terms for \( c^2d^3 \) terms

\( 5c^2d^3 \) (no other like terms, so remains \( 5c^2d^3 \))

Step6: Combine like terms for \( cd^4 \) terms

\( -4cd^4 - 8cd^4 = -12cd^4 \)

Step7: Combine like terms for constant terms

\( 9 - 1 = 8 \)

Step8: Put all combined terms together

\( 6d^5 - 2c^3d^2 + 5c^2d^3 - 12cd^4 + 8 \)

Answer:

A. \( 6d^5 - 2c^3d^2 + 5c^2d^3 - 12cd^4 + 8 \)