Final answer:
To find the term that can create a polynomial in standard form, it must have a degree higher than 6. Options C (3r⁴s⁵) and D (-r⁴s⁶) have degrees 9 and 10 respectively, making them suitable first terms for the expression. The correct answer is option c.
Step-by-step explanation:
The student has asked which term could be used as the first term of the expression to create a polynomial written in standard form. The given expression is + 8r²s⁴ - 3r³s³. In standard form for polynomials, the terms are usually arranged in descending order of their degree (the sum of the exponents of the variables in each term). The degree of the first given term, 8r²s⁴, is 2+4=6, and the degree of the second term, -3r³s³, is 3+3=6 as well. Therefore, the first term of the expression must have a degree higher than 6 to be in standard form.
Let's assess the given options:
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- A. 5s⁷/6: This term has a degree of 7/6, which is not an integer and therefore doesn't fit the criteria for a term in a polynomial.
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- B. s⁵: This term has a degree of 5, which is lower than 6 and thus not suitable.
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- C. 3r⁴s⁵: This term has a degree of 4+5=9, which makes it a suitable first term.
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- D. -r⁴s⁶: This term has a degree of 4+6=10, also suitable.
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- E. -6rs⁵: This term has a degree of 1+5=6, which equals the degree of the existing terms and thus is not suitable.
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- F. 4r/6s: This term does not even have integer exponents, hence it is not a valid term for a polynomial expression.
The correct options for the first term, which would create a polynomial in standard form when added to the given expression, are C. 3r⁴s⁵ and D. -r⁴s⁶.