i don't understand this math problem?could someone help please

Question

Factor the following trinomial.


3r² +8r-3

 

Answer 1

3r²+8r-3 has the general form of cr²+(a+b)r+ab

 

since we are used to solving equations of the form (r+a)(r+b)=r²+(a+b)r+ab let's see if we can make

our equation look like an equation with nothing in front of the r²

Hmm - what could we do? Let's try factoring a 3 from each of the terms

 

3(r²+(8/3)r-1) - we now have an equation of the form r2+(a+b)r+ab - see it?

 

a+b=8/3 and ab=-1 so a=-1/b

 

inserting a=-1/b into a+b=8/3 we get b-1/b=8/3. Now here is a trick to remember if I multiply b by 1 nothing changes  -right? yes because 1xb=b

Sometimes it's easier to see if it's written out.

 

1xb-1/b=8/3  and remember any number divided by itself is =1 for instance b/b=1 - how convenient :)

 

so 1xb-1/b=8/3 can also be written as b(b/b)-1/b=8/3 or (b²-1)/b= 8/3

so the numerators and denominators on both sides of the equation must be equal

from the numerators

b²-1=8

from the denominators

b=3

now solve for a

a+3=8/3 or a=8/3-3=8/3-9/3=-1/3

 

so the solution to the original equation is

 

3(r-1/3)(r+3)=(3r-1)(r+3)

 

Summary steps for a problem where there is a constant in front of the r² (like 3r² in this problem)

1) factor out the constant c in front of the r² from each term - 3(r²+(8/3)r-1) in this problem

2) set a+b=the constant in front of the r -> 8/3 and ab=the constant with no r in it (ab=-1 in this problem)

3) solve for a in terms of b using the ab term (a=-1/b in this problem)

4) insert solution for a into a+b equation  [ (b-1/b) =(b²-1)/b =8/3 in this problem]

           set numerators equal to each other, or denominators equal to each other

           [(b²-1) =8 from the numerators, b=3 from the denominators]

5) solve for a

6) insert a and b into c(r+a)(r+b) [3(r-1/3)(r+3) and get rid of the fraction if necessary=(3r-1)(r+3)]

Answer 2

(3r - 1)(r + 3)

 

Observe that 3r times 3 is 9r, from the O part of FOIL and -1 times r, from the I part, gives middle term +8r when combined.  A good trick to remember!