document.write( "Question 93008: x^2-x-20 \r
\n" ); document.write( "\n" ); document.write( "I need to know how to find the factor of this polynominal? \n" ); document.write( "

Algebra.Com's Answer #67689 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"x%5E2-x-20\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-1\"$ , and the last term is $\"-20\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-20\"$ to get $\"%281%29%28-20%29=-20\"$ .

Now the question is: what two whole numbers multiply to $\"-20\"$ (the previous product) and add to the second coefficient $\"-1\"$ ?

To find these two numbers, we need to list all of the factors of $\"-20\"$ (the previous product).

Factors of $\"-20\"$ :

1,2,4,5,10,20

-1,-2,-4,-5,-10,-20

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to $\"-20\"$ .

1*(-20) = -20
2*(-10) = -20
4*(-5) = -20
(-1)*(20) = -20
(-2)*(10) = -20
(-4)*(5) = -20

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-1\"$ :

\n" ); document.write( "

First Number	Second Number	Sum
1	-20	1+(-20)=-19
2	-10	2+(-10)=-8
4	-5	4+(-5)=-1
-1	20	-1+20=19
-2	10	-2+10=8
-4	5	-4+5=1

From the table, we can see that the two numbers $\"4\"$ and $\"-5\"$ add to $\"-1\"$ (the middle coefficient).

So the two numbers $\"4\"$ and $\"-5\"$ both multiply to $\"-20\"$ and add to $\"-1\"$

Now replace the middle term $\"-1x\"$ with $\"4x-5x\"$ . Remember, $\"4\"$ and $\"-5\"$ add to $\"-1\"$ . So this shows us that $\"4x-5x=-1x\"$ .

$\"x%5E2%2Bhighlight%284x-5x%29-20\"$ Replace the second term $\"-1x\"$ with $\"4x-5x\"$ .

$\"%28x%5E2%2B4x%29%2B%28-5x-20%29\"$ Group the terms into two pairs.

$\"x%28x%2B4%29%2B%28-5x-20%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%28x%2B4%29-5%28x%2B4%29\"$ Factor out $\"5\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

$\"%28x-5%29%28x%2B4%29\"$ Combine like terms. Or factor out the common term $\"x%2B4\"$

===============================================================

Answer:

So $\"x%5E2-x-20\"$ factors to $\"%28x-5%29%28x%2B4%29\"$ .

In other words, $\"x%5E2-x-20=%28x-5%29%28x%2B4%29\"$ .

Note: you can check the answer by expanding $\"%28x-5%29%28x%2B4%29\"$ to get $\"x%5E2-x-20\"$ or by graphing the original expression and the answer (the two graphs should be identical).

\n" ); document.write( " \n" ); document.write( "

\n" );