Question

If p(x)=x^3-3x^2-x+3 and p(3)=0, what is a factor of p(x)

Answer 1

To find a factor of p(x) when p(3) = 0, we can use synthetic division. Dividing p(x) by x - 3 gives a remainder of 0, indicating that x - 3 is a factor.

Step-by-step explanation:

To find a factor of p(x) when p(3) = 0, we can use synthetic division. Since p(3) = 0, it means that x = 3 is a root of the polynomial which implies that x - 3 is a factor.

Using synthetic division, we divide p(x) by x - 3 to find the quotient. The remainder should be 0 if x - 3 is a factor.

Performing synthetic division:

3 | 1 -3 -1 3
|______ 3 0 -3
1 0 -1 0

The remainder is 0, so x - 3 is a factor of p(x).

Answer 2

Answer: x-3

Since p(3) = 0, this means x = 3 plugs into p(x) to get 0

We can write p(x) as p(x) = (x-3)q(x) where q(x) is some other polynomial that multiplies with (x-3) to lead to x^3-3x^2-x+3

Let's plug in x = 3 and see what happens

p(x) = (x-3)q(x)

p(3) = (3-3)q(3)

p(3) = 0*q(3)

p(3) = 0

No matter what the result of q(3) was, it doesn't matter because it multiplies with 0 to get 0.

The general rule is: if p(k) = 0, then x-k is a factor of p(x). This is a special case of the remainder theorem.