Question

If f(x)=x squared +4x-5
show that (x-3) is a factor of the polynomial h(x)=f(x)-f(3)

Answer 1

To show that (x-3) is a factor of the polynomial h(x) = f(x) - f(3), we need to use the definition of a polynomial factor. A polynomial factor is a polynomial that, when multiplied by another polynomial, results in the original polynomial.

First, let's evaluate f(3) to find the value of h(3).

h(3) = f(3) - f(3) = 0

Now let's examine the value of h(x) for any x other than 3.

h(x) = f(x) - f(3)

If (x-3) is a factor of h(x), then for any value of x other than 3, h(x) should be equal to zero.

So,

h(x) = (x-3)g(x)

where g(x) is another polynomial.

Therefore, by definition, (x-3) is a factor of h(x) = f(x) - f(3).