What is the reminder if ax^3+bx^2+cx+d is divided by x-2 ?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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Here we use the remainder theorem. As ax^3 + bx^2 + cx + d is divided by (x-2).

When f(x) is divided by (x-a) the remainder is given by f(a).

So here we have the remainder as f(x) = ax^3 + bx^2 + cx + d for x = 2.

=> a*2^3 + b*2^2 + c*2 + d

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

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

Therefore the remainder when ax^3 + bx^2 + cx + d is divided by (x-2) is 8a + 4b + 2c +d.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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Since the polynomial is divided by the binomial x-2, the reminder is a constant.

We'll write the division with reminder:

ax^3+bx^2+cx+d = (x-2)(ex^2 + fx + g) + h

The reminder R(x) = h

The fundamental theorem of algebra states that the reminder of a polynomial divided by a binomial x-a is:

P(a) = R(a)

We'll substitute x by 2 in the expression of polynomial:

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

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

The reminder is:

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

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