Just a+b=?
If is a perfect square polynomial, where and are real numbers, then what is the values of
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1 solution
If the above polynomial is a perfect square, then it can be written as:
x 4 + 2 x 3 + a x 2 + b x + 9 = ( x 2 + α x + β ) 2 = x 4 + 2 α x 3 + ( 2 β + α 2 ) x 2 + 2 α β x + β 2 ;
which after matching coefficients, we obtain α = 1 , β = ± 3 . These values in turn will yield:
a = 2 ( 3 ) + 1 2 = 7 , b = 2 ( 3 ) ( 1 ) = 6 ⇒ a + b = 1 3 ;
a = 2 ( − 3 ) + 1 2 = − 5 , b = 2 ( − 3 ) ( 1 ) = − 6 ⇒ a + b = − 1 1
of which the second choice makes the cut. Thus the polynomial in question is ( x 2 + x − 3 ) 2 .