what is the polynomial f(x)=x^3-x^2+ax+b if f(0)=4 and x=2 is the root of f?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to determine f(x)=x^3 - x^2 + ax + b  given that f(0)= 4 and x=2 is the root of f(x)

f(0) = 0 - 0 + 0 + b = 4

=> b = 4

f(x) = x^3 - x^2 + ax + 4

x = 2 is a root of f(x)

=> 2^3 - 2^2 + 2a + 4 = 0

=> 8 - 4 + 2a + 4 = 0

=> 2a + 8 = 0

=> a = -4

The function f(x) = x^3 - x^2 - 4x + 4

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll recall the reminder theorem that states the followings;

f(a) = 0 <=> a is the root of the polynomial

f(a) = r, where r is the reminder of polynomial.

According to all the above, we'll have:

f(2) = 0 => 2^3 - 2^2 + 2a + b = 0

8 - 4 + 2a + b = 0

2a + b = -4 (1)

f(0) = 4 => b = 4 (2)

We'll replace (2) in (1):

2a + 4 = -4

2a = -8 => a = -4

The requested polynomial is: f(x) = x^3 - x^2 - 4x + 4.

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