Can you help me?

Question

Im learning about the factor theorem and i cant figure out these two things...

Given a polynomial and one of its factors, find the remaining factors of the polynomials. Some factors may not be binomials.

1.) 2x^3-5x^2-28x+15; x-5

Find values of K ao that each remainder is 3

2.) (x^2+5x+7)/(x+k)

Answer 1

I believe, that this formula will help you too in the future: "ax^2+bx+c=a(x-x‚)(x-x„),  x‚ and x„ are roots of equation ax^2+bx+c=0

Answer 2

For the first, divide 2x³ - 5x² -28x + 15 by (x - 5).  You should get 2x² + 5x - 3.  Now you can factor that:  look for combinations of 1, 2, and 3 that can add to 5 (for 5x) and multiply to get -3.  Let's try (2x - 1) and (x + 3),

since (-1)(+3) = -3 and (2x)(+3) + (x)(-1) = +6x - x = +5x.  So now we have 3 factors of the original problem:  (x - 5)(x + 3)(2x - 1).

For the second:  It's asking what values of k in (x + k) would give you a remainder of +3 after you divide (x² + 5x + 7) by that (x + k).  So let's subtract that 3 from (x² + 5x + 7) and see if what remains can be factored.  (x² + 5x + 7) - 3 = (x² + 5x + 4).  Looking for factors that will multiply to a product of 4 and also add to 5 (for 5x), let's try 4 and 1:  (x + 4)(x + 1) = (x2 + 5x + 4), leaving our remainder of +3.  So our k values are +1 and +4, or (x + 1) and (x + 4).