When x^5, x+1x and 1+2x+3x^3 are multiplied, the product is a polynomial of degree

1983 AMC 12 Problem 6

Attempt Problem 6 of the 1983 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1983 AMC 12 solutions, or check the answer key.

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When

x^5,

x+\frac1x

and

1+\frac2x+\frac3{x^3}

are multiplied, the product is a polynomial of degree

2

3

6

7

8

Track the largest exponent produced by choosing one term from each factor

The leading term comes from

x^5\cdot x\cdot1

The greatest exponent comes from

x^5\cdot x\cdot1=x^6.

Every other term has smaller exponent, and the smallest possible exponent is

5-1-3=1,

so the product is indeed a polynomial. Its degree is

6.

Therefore, the correct answer is C.