The Rational Roots Theorem — Quiz

Question 3 • Mathematics III

According to the Rational Root Theorem, what are all the potential rational roots of f of x is equal to 9 x to the 4th power minus 2 x squared minus 3 x plus 4 ?

Answer

1 fourth , 1 half , 3 fourths , 1 , 3 halves , 9 fourths , 3 , 9 halves , 9

plus or minus 1 ninth , plus or minus 2 ninths , plus or minus 1 third , plus or minus 4 ninths , plus or minus 2 thirds , plus or minus 1 , plus or minus 4 thirds , plus or minus 2 , plus or minus 4

plus or minus 1 fourth , plus or minus 1 half , plus or minus 3 fourths , plus or minus 1 , plus or minus 3 halves , plus or minus 9 fourths , plus or minus 3 , plus or minus 9 halves , plus or minus 9

1 ninth , 2 ninths , 1 third , 4 ninths , 2 thirds , 1 , 4 thirds , 2 , 4

Explanation

The Rational Root Theorem says any rational root must be ±p/q, where p divides the constant term 4 and q divides the leading coefficient 9. Combining the factors of 4 (1, 2, 4) with the factors of 9 (1, 3, 9), then removing duplicates, gives ±1/9, ±2/9, ±4/9, ±1/3, ±2/3, ±4/3, ±1, ±2, and ±4. This is choice D.

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