If a*x^3 + b*x^2 + c*x + d is divided by (x - 2), then what is the remainder?

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sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember the reminder theorem such that:

`f(x) = (x-2)*q(x) + r(x)`

q(x) and r(x) express the quotient and reminder you'll get after division.

You should substitute 2 for x in equation above such that:

`f(2) = (2-2)*q(2) + r(2)`

Notice that you may evaluate f(2) substituting 2 for x in equation of polynomial such that:

`f(2) = 8a + 4b + 2c + d`

You need to substitute`8a + 4b + 2c + d ` for f(2) in equation `f(2) = (2-2)*q(2) + r(2)`  such that:

`8a + 4b + 2c + d = 0*q(2) + r(2)`

`r(2) = 8a + 4b + 2c + d`

Hence, evaluating the reminder r(x) at x=2, under given conditions, yields `r(2) = 8a + 4b + 2c + d` .

justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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The remainder when f(x) = ax^3 + bx^2 + cx + d is divided by x - 2 can be calculated by the remainder theorem. According to this theorem the remainder when a function f(x) is divided by (x - a) is given by f(a)

The required remainer is f(2) = a*2^3 + b*2^2 + c*2 + d

=> 8a + 4b + 2c + d

The remainder of dividing f(x) = ax^3 + bx^2 + cx + d by (x - 2) is 8a + 4b + 2c + d

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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To determine the reminder we'll apply the reminder theorem. We know that the polynomial P(x)= ax3 + bx2 + cx + d at x=2 is the same as the reminder we'll get when we divide P(x) by x-2.

Therefore, we'll calculate P(2) to determine the reminder:

P(2) = a*2^3 + b*2^2 + c*2 + d

P(2) = 8a + 4b + 2c + d

The reminder that we'll get when ax3 + bx2 + cx + d is divided by x - 2 is P(2) = 8a + 4b + 2c + d.

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