Amazing Polynomial!
The polynomial P ( x ) = x 3 + a x 2 + b x + c has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the y -intercept of the graph of y = P ( x ) is 2, then what is the value of − b ?
The answer is 11.
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2 solutions
Damn my silly arithmetic mistakes! :(
Condição I:
3 x ′ + x ′ ′ + x ′ ′ ′ = x ′ ⋅ x ′ ′ ⋅ x ′ ′ ′ = 1 + a + b + c
Condição II:
P ( 0 ) = 2
Por conseguinte, c = 2
Das relações de Girard,
x ′ + x ′ ′ + x ′ ′ ′ = − 1 a
E,
x ′ ⋅ x ′ ′ ⋅ x ′ ′ ′ = − 1 c
Substituindo-as na condição I:
⎩ ⎪ ⎨ ⎪ ⎧ 3 − a = − 2 − 2 = 1 + a + b + 2
Encontremos "a",
− a = − 2 ⋅ 3 a = 6
Por fim,
− 2 = 3 + a + b − 2 = 3 + 6 + b − b = 1 1
mean of it's zeros = - a/3 = the product of it's zeros = - c =sum of its coefficients = 1+a+b+c.
y-intercept of the graph means x=0. So P(0)= 2 =c.
Means - 2 = - c = - a/3 = 1+a+b+c. Solving, -b = 11.