Can the value of k for which 3x^3 + 4x^2 + 7x + k is divisible by (x - 4) be determined without using the remainder theorem.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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Without using the remainder theorem the value of k that makes 3x^3 + 4x^2 + 7x + k divisible by (x - 4) be determined by dividing 3x^3 + 4x^2 + 7x + k by x - 4 and value of k is got by equating the remainder to 0 and solving for k.

(x - 4)| 3x^3 + 4x^2 + 7x + k | 3x^2 + 16x + 71

______3x^3 - 12x^2

--------------------------------------------

_____________16x^2 + 7x + k

_____________16x^2 - 64x

----------------------------------------------

___________________71x + k

___________________71x + -284

-----------------------------------------------

________________________k + 284

 

The required value of k is -284.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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The problem provides the information that x - 4 divides the polynomial `3x^3 + 4x^2 + 7x + k` , hence x - 4 is a factor of polynomial.

You need to use an alternative method to find the value of k, hence you should write the factored form of polynomial such that:

`3x^3 + 4x^2 + 7x + k = (x-4)(ax^2+ bx + c)`

Opening the brackets to the right side yields:

`3x^3 + 4x^2 + 7x + k = ax^3 + bx^2 + cx - 4ax^2 - 4bx - 4c`

Collecting like terms to the right side yields:

`3x^3 + 4x^2 + 7x + k = ax^3 + x^2*(b - 4a) + x*(c - 4b) - 4c`

Equating the coefficients of like powers yields:

`a = 3`

`b - 4a = 4 =gt b - 12 = 4 =gt b = 16`

`c - 4b = 7 =gt c = 7 + 4*16 =gt c = 71`

`-4c = k =gt -4*71 = k =gt k = -284`

Hence, evlauating k, without using reminder theorem, yields k = -284.

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