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What is the right choice for question 16) ? http://postimg.org/image/ksqk61t65/

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lkballer24 | Student, Grade 11 | Valedictorian

Posted August 20, 2013 at 1:22 PM via web

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What is the right choice for question 16) ?

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llltkl | College Teacher | Valedictorian

Posted August 20, 2013 at 1:43 PM (Answer #1)

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A polynomial `f(x)` when divided by a linear polynomial `x-a` , its remainder is given by `f(a)` .

Here, `4x^5-3x^2-4x+1` is divided by `x+1` . so, its remainder is given by f(-1).

`f(-1)=4*(-1)^5-3*(-1)^2-4*(-1)+1=-4-3+4+1=-2`

Therefore, the right choice is option a).

Sources:

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted August 20, 2013 at 4:46 PM (Answer #2)

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You also may use the undetermined coefficients method as alternative method, such that:

- write the reminder theorem using the fact that the reminder is a constant since the divisor is a linear binomial

`4x^5 - 3x^2 - 4x + 1 = (x + 1)(ax^4 + bx^3 + cx^2 + dx + e) + f`

`4x^5 - 3x^2 - 4x + 1 = ax^5 + bx^4 + cx^3 + dx^2 + ex + ax^4 + bx^3 + cx^2 + dx + e + f`

Grouping the terms yields:

`4x^5 - 3x^2 - 4x + 1 = ax^5 + x^4(b + a) + x^3(c + b) + x^2(d + c) + x(e + d) + e + f`

Equating the coefficients of like powers, yields:

`a = 4`

`b + a = 0 => b = -a => b = -4`

`c + b = 0 => c = -b => c = 4`

`d + c = -3 => d + 4 = -3 => d = -7`

`e + d = -4 => e - 7 = -4 => e = 3`

`e + f = 1 => f = 1 - 3 => f = -2`

Hence, since the reminder is f, yields that `f = -2` and you need to validate the answer `a)-2.`

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