Question

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

Explanation:

Step1: Set up subtraction

To find the other addend, subtract the given addend from the total sum:
$$(8d^5 - 3c^3d^2 + 5c^2d^3 - 4cd^4 + 9) - (2d^5 - c^3d^2 + 8cd^4 + 1)$$

Step2: Distribute the negative sign

Apply the negative sign to each term in 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 $d^5$

Subtract coefficients of $d^5$:
$$8d^5 - 2d^5 = 6d^5$$

Step4: Combine like terms for $c^3d^2$

Add coefficients of $c^3d^2$:
$$-3c^3d^2 + c^3d^2 = -2c^3d^2$$

Step5: Combine like terms for $c^2d^3$

There is only one $c^2d^3$ term:
$$5c^2d^3$$

Step6: Combine like terms for $cd^4$

Add coefficients of $cd^4$:
$$-4cd^4 - 8cd^4 = -12cd^4$$

Step7: Combine constant terms

Subtract the constants:
$$9 - 1 = 8$$

Step8: Assemble the polynomial

Combine all simplified terms:
$$6d^5 - 2c^3d^2 + 5c^2d^3 - 12cd^4 + 8$$

Answer:

6d⁵ - 2c³d² + 5c²d³ - 12cd⁴ + 8