find the value of p n q if x^3+px^2-qx+30 has (x-2) and (x-3) as its 2 factors. find also the third factor?

Answer 1

Your polynome translates into (x-2)(x-3)(x-a) where (x-a) is the unknown factor
(x-2)(x-3)(x-a) = (x^2-5x+6)(x-a) =
x^3 - x^2(a+5) + x( 5a+6) - 6a
so p = -(a+5)
q=5a+6 and 30 = -6a
a= -5
q= -19
p=0

Answer 2

The first poster outlined the simplest way to do things. However, you can also divide:

x ^ 3 + px ^ 2 - qx + 30 / x - 2 = x ^ 2 + (p-2)x + (-q + 2p - 4)
x ^ 3 - 2x ^ 2
-----------------
(p-2)x ^ 2 - qx
(p-2)x ^ 2 - (2p-4)x
-------------------------
(-q + 2p - 4)x + 30
(-q + 2p - 4)x + -2q + 4p - 8
------------------------------------
30 + 2q - 4p + 8

So now you know that, since there is no remainder:

30 + 2q - 4p + 8 = 0
2q - 4p = -38
2p - q = 19

Now you do the second division:

x ^ 3 + px ^ 2 - qx + 30 / x - 3 = x ^ 2 + (p-3)x + (-qx + 3p - 9)
x ^ 3 - 3x ^ 2
-----------------
(p-3)x ^ 2 - qx
(p-3)x ^ 2 - (3p-9)x
--------------------------
(-qx + 3p - 9)x + 30
(-qx + 3p - 9)x + (-3q + 9p - 27)
-----------------------------------------
30 + 3q - 9p + 27

So now you now that:

30 + 3q - 9p + 27 = 0
3q - 9p = - 57
q - 3p = -19

Combining the two equations you have:

2p - q = 19
q - 3p = -19

q = 2p - 19
(2p - 19) - 3p = -19
p = 0

2(0) - q = -19
q = -19

So now you know that p is 0 and q is -19. It's very easy to figure out the third factor at this point.

Answer 3

The easiest way to figure this out is as follows:

You know the constant in the equation is 30.
(-2)*(-3) is +6.
30/6 = 5 so the last factor is (x+5). From here, just expand (x-2)(x-3)(x+5) and you'll see what p and q are.

Answer 4

p=2^ q=3^

Third factor=(x-2)

10+2+1+30=43

Ten Two One Thirty out of Fourty-Three it's bad luck Thirteen!!!

Answer 5

44

Answer 6

(x - 2)(x - 3)(x + n)

(x^2 - 3x - 2x + 6)(x + n)
(x^2 - 5x + 6)(x + n)
x^3 + nx^2 - 5x^2 - 5nx + 6x + 6n
x^3 + (n - 5)x^2 + (-5n + 6)x + 6n
x^3 + (n - 5)x^2 - (5n - 6)x + 6n

6n = 30
n = 5

p = n - 5
p = 5 - 5
p = 0

q = 5n - 6
q = 5(5) - 6
q = 25 - 6
q = 19

p = 0
q = 19
Third factor : (x + 5)