document.write( "Question 173192This question is from textbook Introductory Algebra
\n" ); document.write( ": Factor completely\r
\n" ); document.write( "\n" ); document.write( "4x^3 + 4x^2 - 80x \n" ); document.write( "

Algebra.Com's Answer #128064 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( " $\"4x%5E3%2B4x%5E2-80x\"$ Start with the given expression\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"4x%28x%5E2%2Bx-20%29\"$ Factor out the GCF $\"4x\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"x%5E2%2Bx-20\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "------------------------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Looking at $\"1x%5E2%2B1x-20\"$ we can see that the first term is $\"1x%5E2\"$ and the last term is $\"-20\"$ where the coefficients are 1 and -20 respectively.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -20 to get -20. Now what two numbers multiply to -20 and add to the middle coefficient 1? Let's list all of the factors of -20:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of -20:\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,10,20\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-10,-20 ...List the negative factors as well. This will allow us to find all possible combinations\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to -20\r
\n" ); document.write( "\n" ); document.write( "(1)*(-20)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-10)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-5)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(20)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(10)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(5)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "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\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\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

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From this list we can see that -4 and 5 add up to 1 and multiply to -20\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"1x%5E2%2B1x-20\"$ , replace $\"1x\"$ with $\"-4x%2B5x\"$ (notice $\"-4x%2B5x\"$ adds up to $\"1x\"$ . So it is equivalent to $\"1x\"$ )\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"1x%5E2%2Bhighlight%28-4x%2B5x%29%2B-20\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"1x%5E2-4x%2B5x-20\"$ by grouping:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%281x%5E2-4x%29%2B%285x-20%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x-4%29%2B5%28x-4%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"5\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%2B5%29%28x-4%29\"$ Since we have a common term of $\"x-4\"$ , we can combine like terms\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"1x%5E2-4x%2B5x-20\"$ factors to $\"%28x%2B5%29%28x-4%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this also means that $\"1x%5E2%2B1x-20\"$ factors to $\"%28x%2B5%29%28x-4%29\"$ (since $\"1x%5E2%2B1x-20\"$ is equivalent to $\"1x%5E2-4x%2B5x-20\"$ )\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "------------------------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"4x%28x%5E2%2Bx-20%29\"$ and factors further to $\"4x%28x%2B5%29%28x-4%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "------------------\r
\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"4x%5E3%2B4x%5E2-80x\"$ factors to $\"4x%28x%2B5%29%28x-4%29\"$ \r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );