Find value of k in (x+1)(x^2+kx-4)=x^3+4x^2-x-4?

Question

Could you show me how to complete this one..step by step..it's been about 20 years for me and trying to help my daughter is tough!!! HELP!!

Others are... (2x-1)(x^2+kx+3)=2x^3-5x^2+8x-3 and
(3x+1)(x^2+kx+1)=3x^3-5x^2+x+1

Answer 1

4x^2 is the key.

You get 1 x^2 by multiplying 1 by x^2, so kx times x must equal 4-1=3.

In the same way for the second:

You get -1 x^2 by multiplying -1 by x, so 2*k*x^2 = -5-(-1)x^2

2*k*x^2 = -4x^2
k=-4/2
k=-2.

The second is similar. Basically, look at the x^2 coefficient and look at how many you have through the terms NOT including k, then subtract the number of other ones from the total number of x^2. kx must be multiplied by x to get x^2, so take into account the other x coefficient.

i.e. (2x)(kx) = 2kx^2

Answer 2

Hi, Remember that k is simply a coefficient to the variable x. It looks like multiplying the factors on the left side is first. yielding x^3+kx^2-4x+x^2+kx-4=x^3+4x^2-x-4
then simplify by bringing the opposite of the term over to the left side
x^3-x^3+x^2-4x^2+kx^2-4x+kx+x-4+4 which yields
-3kx^2+2kx
which can then be factored to -k(3x^2-2x)=0
If this is insufficient for an answer let me know and i can send you a more detailed answer using my computer algebra system which does a beautiful job of simplifying and formating.

Answer 3

The x² terms are going to come from the x*kx and 1*x² in the original product, so the x² term will have a coefficient of k+1. Since the given product has an x² coefficient of 4, then k must be 3.


Doug

Answer 4

(x+1)(x^2+kx-4) = x^3 + 4x^2 -x-4

multiple out the left hand side

x^3 + kx^2 -4x + x^2 +kx -4 = x^3 + 4x^2 -x-4

pull together similar terms on the left side

x^3 + (k+1)x^2 + (k - 4)x - 4 = x^3 + 4x^2 - x - 4

cancel out terms that are the same on both sides

(k+1)x^2 + (k - 4)x = 4x^2 - x

thus you can easily see the answer is k=3 without any more calculations. If the other answers are not as apparent continue to divide out similar terms from both sides .

Answer 5

If I can use the last problem posed as an example. . .

since both K and X are unknown variables the answer is not likely to be a concrete value. multiply and simplify until you can solve for "K":

(3x+1)(x^2 +kx+1)=3x^3-5x^2 +x+1
=> 3kx^2 +2x +6x^2 +kx = 0
=> 3kx^2 +kx = -6x^2-2x
=> 3kx+k = -6x-2
=> k(3x+1) = -6x+2
=> k= (-6x+2)/(3x+1)