Question

what is the difference of the polynomials?
$(8r^{6}s^{3} - 9r^{5}s^{4} + 3r^{4}s^{5}) - (2r^{4}s^{5} - 5r^{3}s^{6} - 4r^{5}s^{4})$
\bigcirc $6r^{6}s^{3} - 4r^{5}s^{4} + 7r^{4}s^{5}$
\bigcirc $6r^{6}s^{3} - 13r^{5}s^{4} - r^{4}s^{5}$
\bigcirc $8r^{6}s^{3} - 5r^{5}s^{4} + r^{4}s^{5} + 5r^{3}s^{6}$
\bigcirc $8r^{6}s^{3} - 13r^{5}s^{4} + r^{4}s^{5} - 5r^{3}s^{6}$

Explanation:

Step1: Distribute the negative sign

To find the difference of the polynomials, we first distribute the negative sign to the second polynomial:
$$(8r^{6}s^{3}-9r^{5}s^{4}+3r^{4}s^{5})-(2r^{4}s^{5}-5r^{3}s^{6}-4r^{5}s^{4}) = 8r^{6}s^{3}-9r^{5}s^{4}+3r^{4}s^{5}-2r^{4}s^{5}+5r^{3}s^{6}+4r^{5}s^{4}$$

Step2: Combine like terms

• For the \(r^{6}s^{3}\) term: There is only \(8r^{6}s^{3}\), so it remains as is.
• For the \(r^{5}s^{4}\) terms: \(-9r^{5}s^{4}+4r^{5}s^{4}=-5r^{5}s^{4}\) (wait, no, wait, let's recalculate: \(-9r^{5}s^{4}+4r^{5}s^{4}=(-9 + 4)r^{5}s^{4}=-5r^{5}s^{4}\)? Wait, no, original second polynomial after distributing the negative: \(-9r^{5}s^{4}+4r^{5}s^{4}\)? Wait, no, the original first polynomial has \(-9r^{5}s^{4}\), and the second polynomial after distributing the negative has \(+4r^{5}s^{4}\) (because the second polynomial's term is \(-4r^{5}s^{4}\), so distributing the negative gives \(+4r^{5}s^{4}\)). So \(-9r^{5}s^{4}+4r^{5}s^{4}=(-9 + 4)r^{5}s^{4}=-5r^{5}s^{4}\)? Wait, no, wait, let's check again. Wait, the first polynomial: \(8r^{6}s^{3}-9r^{5}s^{4}+3r^{4}s^{5}\)

The second polynomial after distributing the negative: \(-2r^{4}s^{5}+5r^{3}s^{6}+4r^{5}s^{4}\)
Now combine like terms:

• \(r^{6}s^{3}\): \(8r^{6}s^{3}\)
• \(r^{5}s^{4}\): \(-9r^{5}s^{4}+4r^{5}s^{4}=(-9 + 4)r^{5}s^{4}=-5r^{5}s^{4}\)
• \(r^{4}s^{5}\): \(3r^{4}s^{5}-2r^{4}s^{5}=(3 - 2)r^{4}s^{5}=r^{4}s^{5}\)
• \(r^{3}s^{6}\): \(5r^{3}s^{6}\)

So putting it all together:
$$8r^{6}s^{3}-5r^{5}s^{4}+r^{4}s^{5}+5r^{3}s^{6}$$ Wait, but let's check the options. Wait, option C is \(8r^{6}s^{3}-5r^{5}s^{4}+r^{4}s^{5}+5r^{3}s^{6}\), which matches our result. Wait, but let's re - check the combination of \(r^{5}s^{4}\) terms: \(-9r^{5}s^{4}+4r^{5}s^{4}\). \(-9 + 4=-5\), so \(-5r^{5}s^{4}\). \(r^{4}s^{5}\) terms: \(3r^{4}s^{5}-2r^{4}s^{5}=r^{4}s^{5}\). The \(r^{6}s^{3}\) term is \(8r^{6}s^{3}\) and the \(r^{3}s^{6}\) term is \(5r^{3}s^{6}\). So the polynomial after combining like terms is \(8r^{6}s^{3}-5r^{5}s^{4}+r^{4}s^{5}+5r^{3}s^{6}\)

Answer:

C. \(8r^{6}s^{3}-5r^{5}s^{4}+r^{4}s^{5}+5r^{3}s^{6}\)