Question

Determine m if the polynomial P is divisible by x-3. P=x^3+mx^2+3mx-9

Accepted Answer

P= x^3 + mx^2 + 3mx -9
P is dividible by (x-3)
Then (x-3) is one of P's factors, and 3 is one of the roots for P.
==> P(x) = (x-3)*Q(x)
==> P(3) = 0
P(3) = 3^3 + m(3^2) + 3m(3) - 9 = 0
==> 27 + 9m + 9m - 9 = 0
==> 18 + 18m = 0
==> 18m = -18
==> m = -1

Answer

P(x) = x^3+mx^2+3mx-9
If P(x)  is divisible by x-3, then by remainder theorem P(3) = 0
So P(3) = 3^3-m*3^2+3m*3-9 = 0
27 - 9m^2+9m -9 = 0. Divide by 9
3-m^2+m-1 = 0
-m^2+m+2 = 0. Multiply by -1.
m^2-m-2 = 0
m^2-2m+m-2 = 0
m(m-2)+1(m-2) = 0
(m-2)(m+1) = 0
m-2 = 0 or m+1 =0
So m=2 or m = -1.

Answer

If the polynomial is divisible by (x-3), then x=3 verify the equation:
P(3)=0
This equation is the result of applying the rule of the division with reminder:
P(x) = (x-3)*Q(x)
The reminder is R=0, because P is divisible by x-3.
Now, we'll substitute x by 3 and we'll get:
P(3) = (3-3)*Q(3)
P(3) = 0
P(3) = 3^3 + m*3^2 + 3*m*3 - 9
P(3) = 27 + 9m + 9m - 9
We'll combine like terms:
18m + 18 = 0
We'll factorize:
18 (m+1) = 0
We'll divide by 18:
m+1 = 0
We'll subtract 1 both sides:
m=-1

Math

Determine m if the polynomial P is divisible by x-3. P=x^3+mx^2+3mx-9

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