Question

1. an expression is shown:

\\(\frac{1}{2}r^6t^2 (2r^3t^2 - r^6t^2 + 8rt^2)\\)
what is the product of the polynomials?
a. \\(r^9t^4 - \frac{1}{2}r^6t^2 + 4r^7t^4\\)
b. \\(4r^{18}t^4 - 2r^{36}t^4 + 16r^6t^4\\)
c. \\(r^9t^4 - \frac{1}{2}r^{12}t^4 + 4r^7t^4\\)
d. \\(r^{19}t^4 - \frac{1}{2}r^{36}t^4 + 4r^6t^4\\)

Explanation:

Step1: Distribute the monomial

We need to multiply \(\frac{1}{2}r^{6}t^{2}\) with each term inside the parentheses \((2r^{3}t^{2}-r^{6}t^{2} + 8rt^{2})\). Using the distributive property \(a(b + c + d)=ab+ac + ad\), we get:

• For the first term: \(\frac{1}{2}r^{6}t^{2}\times2r^{3}t^{2}\)
• For the second term: \(\frac{1}{2}r^{6}t^{2}\times(-r^{6}t^{2})\)
• For the third term: \(\frac{1}{2}r^{6}t^{2}\times8rt^{2}\)

Step2: Simplify each product

• First term: \(\frac{1}{2}\times2\times r^{6 + 3}\times t^{2+2}=r^{9}t^{4}\) (using the rule \(a^{m}\times a^{n}=a^{m + n}\) for exponents)
• Second term: \(\frac{1}{2}\times(- 1)\times r^{6+6}\times t^{2 + 2}=-\frac{1}{2}r^{12}t^{4}\)
• Third term: \(\frac{1}{2}\times8\times r^{6+1}\times t^{2+2}=4r^{7}t^{4}\)

Step3: Combine the simplified terms

Combining the three terms we got from the previous step, we have \(r^{9}t^{4}-\frac{1}{2}r^{12}t^{4}+4r^{7}t^{4}\)

Answer:

C. \(r^{9}t^{4}-\frac{1}{2}r^{12}t^{4}+4r^{7}t^{4}\)