We want to completely factorize the given polynomial.
We will find that the factorized polynomial is:
p(x) = (x - 4)*(x - 1)*(x - 3)*(x + 3)
We start with p(x) = (x^2 – 5x + 4)(x^2 – 9) and we want to completely factorize it.
To do so, we need to factorize the two factors.
The right factor is easy to factorize, we have:
x^2 - 9
To factorize it, we need to find the two roots, so we need to solve:
x^2 - 9 = 0
x^2 = 9
x = ± √9 = ±3
Then we have two roots, x = -3 and x = 3
We can write this as:
x^2 - 9 = (x - 3)*(x - (-3)) = (x - 3)*(x + 3)
Now we need to factorize the other part:
x^2 – 5x + 4
To find the roots we will use the Bhaskara's formula, we will find that the roots are:
[tex]x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4*4*1} }{2*1} = \frac{5 \pm3 }{2}[/tex]
Then the two roots are:
x = (5 + 3)/2 = 4
x = (5 - 3)/2 = 1
Then we can write:
x^2 – 5x + 4 = (x - 4)*(x - 1)
Finally, we replace all what we got in the polynomial to get:
p(x) = (x^2 – 5x + 4)(x^2 – 9)
p(x) = (x - 4)*(x - 1)*(x - 3)*(x + 3)
This is the completely factorized polynomial.
If you want to learn more, you can read:
https://brainly.com/question/12787576
