We want to simplify a rational expression, to do it, we need to factorize the numerator.
We need to simplify the expression:
[tex]\frac{p^2 - 4p - 32}{p + 4}[/tex]
We will get:
[tex]\frac{p^2 - 4p - 32}{p + 4} = p - 8[/tex]
To simplify this we need to factorize the numerator. Remember that any square polynomial with roots x₁ and x₂ with a leading coefficient a can be written as:
a*(x - x₁)*(x - x₂)
So we need to find the two solutions of:
[tex]p^2 -4p - 32 = 0[/tex]
To get these we need to use Bhaskara's formula, we will get:
[tex]p = \frac{-(-4) \pm \sqrt{(-4)^2 -4*(-32)*1} }{2*1} \\\\p = \frac{{4 \pm 12} }{2}[/tex]
Then the two solutions are:
- p = (4 + 12)/2 = 8
- p = (4 - 12)/2 = -4
Thus we can write:
[tex]p^2 -4p - 32 = (p + 4)*(p - 8)[/tex]
So we can write the fraction as:
[tex]\frac{p^2 - 4p - 32}{p + 4} = \frac{(p +4)*(p-8)}{p + 4}[/tex]
We can see that we have a factor (p + 4) in both the numerator and denominator, so we can cancel it
[tex]\frac{p^2 - 4p - 32}{p + 4} = \frac{(p +4)*(p-8)}{p + 4} = p - 8[/tex]
p - 8 is the simplification of the fraction, as you can see, there are no restrictions.
If you want to learn more, you can read:
https://brainly.com/question/4053899
