Factor the polynomial 48-20x+2x^2 if the polynomial is not factorable write prime

Question

if the polynomial is not factorable write prime

 

please provide steps thanks

Answer 1

First, write the polynomial in normal order:

 

2x² - 20x + 48

 

We know the factored form looks like this:

 

(2x + a)(x + b)

 

Since the middle term is negative (-20) and last term is positive (48), we know that both a and b must be negative.

This is because a negative + a negative is a negative, and a negative * negative is positive.

 

So we have:

 

(2x - a)(x - b)

 

We know the following:

 

-2bx - ax = -20x

-a * -b = 48

 

If we solve the second equation for a we get:

 

-a = 48 / -b

a = 48 / b

 

Now we can substitute a in the first equation to get:

 

-2bx - (48/b)x = -20x

 

divide both sides by 2x to get:

 

-b - 24/b = -10

-b² - 24 = -10b

b² + 24 = 10b

b² - 10b -24 = 0

 

solve for b:

 

(b - 6)(b - 4) = 0

Therefore b = 6 or b = 4

 

Since a = 48/b, if b = 6 then a = 8. If b = 4 then a = 12.

So we just need to try these values in the original equation to find which is true:

 

The factored equation is:

 

(2x - a)(x - b)

 

if a = 8 and b = 6, we have:

(2x - 8)(x - 6) = 2x² - 12x - 8x + 48 = 2x² - 20x + 48

 

if a=12 and b=4 we have:

(2x - 12)(x - 4) = 2x² - 8x - 12x + 48 = 2x² - 20x + 48

 

Therefore both sets of factors work.