Help with Math Question Needed. Please help...?

(x - k) is a factor of a polynomial P(x) if P(k) = 0

P(x) = kx^3 + 5x^2 - 7kx - 8

P(k) = k(+k)^3 + 5(+k)^2 - 7k(+k) - 8 = k^4 - 2k^2 - 8

P(k) = 0 if k^4 - 2k^2 - 8 = 0

set k^2 = w, so that k^4 = (k^2)^2 = w^2

The equation becomes

w^2 - 2w - 8 = 0

w = - 2, w = 4

substitute back k^2

k^2 = -2 has no real solutions

k^2 = 4

k = ±2

if k = 2

P(x) = 2x^3 + 5x^2 - 14x - 8

synthetic division

__________2______5_______-14_____|__-8

______2__________4________18_____|__8

__________2______9________4______|__0

Quotient Q(x) = 2x^2 + 9x + 4

P(x) = (x - 2)(2x^2 + 9x + 4)

Q(x) can be factored solving

2x^2 + 9x + 4 = 0

x = - 1/2, x = -4

2x^2 + 9x + 4 = 2(x - (-1/2))(x - (-4)) = 2(x + 1/2)(x + 4) = (2x + 1)(x + 4)

putting all together

2x^3 + 5x^2 - 14x - 8 = (x - 2)(2x + 1)(x + 4)

if k = - 2, with a similar procedure you get

P(x) = - 2x^3 + 5x^2 + 14x - 8 = - (x + 2)(x - 4)(2x - 1)

The Factor Theorem states that if x - h is a factor of f(x) then f(h) = 0. In this case, x - k is a factor of f(x) so f(k) = 0. Use Synthetic Division for this - it automatically gives you , first , the value of k, second,the coefficients of the necessary quadratic factor. I will try to set it out, but be ready to rewrite it - "Answers" has a wonderful knack of totally destroying the layout of any sort of table !

k | k 5 -7 -8

k^2 k^3 + 5k k^4 + 5k^2 - 7k

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

k k^2 + 5 k^3 + 5k - 7 k^4 + 5k^2 - 7k - 8 = f(k)

and there we hit an unsurmountable problem - there is no way you can solve this equation f(k) = 0.

Go back to the original question and look for typos I suspect the k in front of the x^3 may (repeat, may) be an error.

Let p(x)=kx^3+5x^2-7kx-8

x-k is a factor=>p(k)=0=>

k^4+5k^2-7k^2-8=0=>

k^4-2k^2-8=0=>

k=+/-2 or k=+/-sqrt(2)i

So,

p(x)=(x-2)(2x^2+9x+4)=>

p(x)=(x-2)(x+4)(2x+1)

or

p(x)=-(x+2)(x-4)(2x-1)

or

p(x)=[x-sqrt(2)i]

[sqrt(2)ix^2+3x-4sqrt(2)i]

or

p(x)=-[x+sqrt(2)i]

[sqrt(2)ix^2-3x-4sqrt(2)i]