When the polynomial P(x) = 3x^3 + kx^2 + 45x + 25 is divided by x + 5, the remainder is 0. Which of the following is also a factor of P(x)?

When the polynomial P(x) = 3x^3 + kx^2 + 45x + 25 is divided by x + 5, the remainder is 0. Which of the following is also a factor of P(x)?

Chris has used the factor theorem.

I will try and walk you through why it works :)

If the remainder is 0 when P(x) is divided by x+5 that must mean that   x+5 is a factor of  P(x)

so     P(x) = (x+5) (some other polynomial)

If this is true then  P(x) must equal zero when   x+5=0     and that happens when  x=-5

so  P(x) must =0  when  x=-5

that is       P(-5)=0

Now I am up to where Chris started.  :)

It will make life easier for you if you work out what I am saying.  The more things you understand the better it is for you :)

If x + 5  is a factor, then -5 is a root.....so we can find the value  of "k"  as follows:

3(-5)^3 + k(-5)^2 + 45(-5) + 25  = 0

-375  + 25k - 225 + 25 = 0

25k = 375 + 225 - 25

25k = 575

k = 23

And using sythetic division, we have

-5   [ 3       23       45       25  ]

                -15      -40       25   

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

       3           8         5        0

So......the  remaining polynomial  is    3x^2 + 8x + 5

Setting this to 0 and factoring, we get :

(3x + 5) (x + 1)  = 0

(C)  is the answer