Let f ( x ) and g ( x ) be two monic cubic polynomials, and let r be a real numb

Annika Miranda

Annika Miranda

Answered question

2022-06-04

Let f ( x ) and g ( x ) be two monic cubic polynomials, and let r be a real number. Two of the roots of f ( x ) are r + 1 and r + 7. Two of the roots of g ( x ) are r + 3 and r + 9 , and
f ( x ) g ( x ) = r
for all real numbers x. Find r

Answer & Explanation

elpartizano7b3mv

elpartizano7b3mv

Beginner2022-06-05Added 3 answers

Your approach is fine. Note that
f ( x ) g ( x ) = = ( 4 p + q ) x 2 + ( 2 r p + 8 p 12 q 2 q r 4 r 20 ) x p r 2 + q r 2 8 p r + 12 q r 7 p + 27 q .
So, 4 p + q = 0; in other words, p = q + 4. Replacing p with q + 4 in the coefficient of x in f ( x ) g ( x ), we get that 4 ( 3 q + r ) = 0; in other words, q = r + 3. And if replace q with r + 3 in the constant term of f ( x ) g ( x ), we get 32. But we want this to be equal to r. Therefore, r = 32

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