SOLUTION: How do I factor ? t (squared) -2t-8 {over} 4(-t squared) {multiplied by} t(squared)-5t+6 {over} t(squared)-t-12

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: How do I factor ? t (squared) -2t-8 {over} 4(-t squared) {multiplied by} t(squared)-5t+6 {over} t(squared)-t-12      Log On


   

Question 78831: How do I factor ?

t (squared) -2t-8 {over} 4(-t squared) {multiplied by} t(squared)-5t+6 {over} t(squared)-t-12
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Does your problem look like this?

%28%28t%5E2-2t-8%29%2F%284-t%5E2%29%29%2A%28%28t%5E2-5t%2B6%29%2F%28t%5E2-t-12%29%29

In order to simplify this algebraic expression, we need to factor each term

So lets factor the first numerator %28t%5E2-2t-8%29

Solved by pluggable solver: Factoring Quadratics with a leading coefficient of 1 (a=1)
In order to factor 1%2Ax%5E2%2B-2%2Ax%2B-8, first we need to ask ourselves: What two numbers multiply to -8 and add to -2? Lets find out by listing all of the possible factors of -8


Factors:

1,2,4,8,

-1,-2,-4,-8,List the negative factors as well. This will allow us to find all possible combinations

These factors pair up to multiply to -8.

(-1)*(8)=-8

(-2)*(4)=-8

Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2

||||
First Number|Second Number|Sum
1|-8|1+(-8)=-7
2|-4|2+(-4)=-2
-1|8|(-1)+8=7
-2|4|(-2)+4=2
We can see from the table that 2 and -4 add to -2.So the two numbers that multiply to -8 and add to -2 are: 2 and -4 Now we substitute these numbers into a and b of the general equation of a product of linear factors which is: %28x%2Ba%29%28x%2Bb%29substitute a=2 and b=-4 So the equation becomes: (x+2)(x-4) Notice that if we foil (x+2)(x-4) we get the quadratic 1%2Ax%5E2%2B-2%2Ax%2B-8 again



Now lets factor the 1st denominator %284-t%5E2%29

Notice this is a difference of squares so we have

%284-t%5E2%29=%282%2Bt%29%282-t%29

Now lets factor the 2nd numerator %28t%5E2-5t%2B6%29
Solved by pluggable solver: Factoring Quadratics with a leading coefficient of 1 (a=1)
In order to factor 1%2Ax%5E2%2B-5%2Ax%2B6, first we need to ask ourselves: What two numbers multiply to 6 and add to -5? Lets find out by listing all of the possible factors of 6


Factors:

1,2,3,6,

-1,-2,-3,-6,List the negative factors as well. This will allow us to find all possible combinations

These factors pair up to multiply to 6.

1*6=6

2*3=6

(-1)*(-6)=6

(-2)*(-3)=6

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5

||||
First Number|Second Number|Sum
1|6|1+6=7
2|3|2+3=5
-1|-6|-1+(-6)=-7
-2|-3|-2+(-3)=-5
We can see from the table that -2 and -3 add to -5.So the two numbers that multiply to 6 and add to -5 are: -2 and -3 Now we substitute these numbers into a and b of the general equation of a product of linear factors which is: %28x%2Ba%29%28x%2Bb%29substitute a=-2 and b=-3 So the equation becomes: (x-2)(x-3) Notice that if we foil (x-2)(x-3) we get the quadratic 1%2Ax%5E2%2B-5%2Ax%2B6 again



Now lets factor the 2nd denominator t%5E2-t-12

Solved by pluggable solver: Factoring Quadratics with a leading coefficient of 1 (a=1)
In order to factor 1%2Ax%5E2%2B-1%2Ax%2B-12, first we need to ask ourselves: What two numbers multiply to -12 and add to -1? Lets find out by listing all of the possible factors of -12


Factors:

1,2,3,4,6,12,

-1,-2,-3,-4,-6,-12,List the negative factors as well. This will allow us to find all possible combinations

These factors pair up to multiply to -12.

(-1)*(12)=-12

(-2)*(6)=-12

(-3)*(4)=-12

Now which of these pairs add to -1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -1

||||||
First Number|Second Number|Sum
1|-12|1+(-12)=-11
2|-6|2+(-6)=-4
3|-4|3+(-4)=-1
-1|12|(-1)+12=11
-2|6|(-2)+6=4
-3|4|(-3)+4=1
We can see from the table that 3 and -4 add to -1.So the two numbers that multiply to -12 and add to -1 are: 3 and -4 Now we substitute these numbers into a and b of the general equation of a product of linear factors which is: %28x%2Ba%29%28x%2Bb%29substitute a=3 and b=-4 So the equation becomes: (x+3)(x-4) Notice that if we foil (x+3)(x-4) we get the quadratic 1%2Ax%5E2%2B-1%2Ax%2B-12 again



So our whole expression factors to




Notice these terms cancel

So we're left with
%28%28t-3%29%28t-2%29%29%2F%28%282-t%29%28t%2B3%29%29
Now factor out a -1 to make t-2 become 2-t

-%28%28t-3%29%282-t%29%29%2F%28%282-t%29%28t%2B3%29%29

-%28%28t-3%29cross%28%282-t%29%29%29%2F%28cross%28%282-t%29%29%28t%2B3%29%29 Notice these terms cancel
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Answer:
So the whole thing reduces to

-%28t-3%29%2F%28t%2B3%29