document.write( "Question 177243: i'm suppose to factor completely and can you please show the steps, thank you\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "61) 12x3 + 16x2 – 400x
\n" ); document.write( "
\n" ); document.write( "62) x3 + 3x2 + 2x + 6
\n" ); document.write( "
\n" ); document.write( "63) x2 – 3x + 2
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #132319 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
# 61\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"12x%5E3%2B16x%5E2-400x\"$ Start with the given expression\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"4x%283x%5E2%2B4x-100%29\"$ Factor out the GCF $\"4x\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"3x%5E2%2B4x-100\"$ \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 $\"3x%5E2%2B4x-100\"$ we can see that the first term is $\"3x%5E2\"$ and the last term is $\"-100\"$ where the coefficients are 3 and -100 respectively.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 3 and the last coefficient -100 to get -300. Now what two numbers multiply to -300 and add to the middle coefficient 4? Let's list all of the factors of -300:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of -300:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,10,12,15,20,25,30,50,60,75,100,150,300\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-25,-30,-50,-60,-75,-100,-150,-300 ...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 -300\r
\n" ); document.write( "\n" ); document.write( "(1)*(-300)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-150)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-100)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-75)\r
\n" ); document.write( "\n" ); document.write( "(5)*(-60)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-50)\r
\n" ); document.write( "\n" ); document.write( "(10)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(12)*(-25)\r
\n" ); document.write( "\n" ); document.write( "(15)*(-20)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(300)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(150)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(100)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(75)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(60)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(50)\r
\n" ); document.write( "\n" ); document.write( "(-10)*(30)\r
\n" ); document.write( "\n" ); document.write( "(-12)*(25)\r
\n" ); document.write( "\n" ); document.write( "(-15)*(20)\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 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	-300	1+(-300)=-299
2	-150	2+(-150)=-148
3	-100	3+(-100)=-97
4	-75	4+(-75)=-71
5	-60	5+(-60)=-55
6	-50	6+(-50)=-44
10	-30	10+(-30)=-20
12	-25	12+(-25)=-13
15	-20	15+(-20)=-5
-1	300	-1+300=299
-2	150	-2+150=148
-3	100	-3+100=97
-4	75	-4+75=71
-5	60	-5+60=55
-6	50	-6+50=44
-10	30	-10+30=20
-12	25	-12+25=13
-15	20	-15+20=5

\r
\n" ); document.write( "\n" ); document.write( "None of these pairs of factors add to 4. So the expression $\"3x%5E2%2B4x-100\"$ cannot be factored\r
\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( "\n" ); document.write( "Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"12x%5E3%2B16x%5E2-400x\"$ factors to $\"4x%283x%5E2%2B4x-100%29\"$ \r
\n" ); document.write( "\n" ); document.write( " \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "# 62\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%5E3%2B3x%5E2%2B2x%2B6\"$ Start with the given expression\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E3%2B3x%5E2%29%2B%282x%2B6%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%5E2%28x%2B3%29%2B2%28x%2B3%29\"$ Factor out the GCF $\"x%5E2\"$ out of the first group. Factor out the GCF $\"2\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B2%29%28x%2B3%29\"$ Since we have the common term $\"x%2B3\"$ , we can combine like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"x%5E3%2B3x%5E2%2B2x%2B6\"$ factors to $\"%28x%5E2%2B2%29%28x%2B3%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E3%2B3x%5E2%2B2x%2B6=%28x%5E2%2B2%29%28x%2B3%29\"$ \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( "# 63\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2-3x%2B2\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-3\"$ , and the last term is $\"2\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"2\"$ to get $\"%281%29%282%29=2\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"2\"$ (the previous product) and add to the second coefficient $\"-3\"$ ?\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"2\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"2\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2\r
\n" ); document.write( "\n" ); document.write( "-1,-2\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"2\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*2
\n" ); document.write( "(-1)*(-2)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-3\"$ :\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	2	1+2=3
-1	-2	-1+(-2)=-3

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-1\"$ and $\"-2\"$ add to $\"-3\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-1\"$ and $\"-2\"$ both multiply to $\"2\"$ and add to $\"-3\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-3x\"$ with $\"-x-2x\"$ . Remember, $\"-1\"$ and $\"-2\"$ add to $\"-3\"$ . So this shows us that $\"-x-2x=-3x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-x-2x%29%2B2\"$ Replace the second term $\"-3x\"$ with $\"-x-2x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-x%29%2B%28-2x%2B2%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x-1%29%2B%28-2x%2B2%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x-1%29-2%28x-1%29\"$ Factor out $\"2\"$ 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.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x-2%29%28x-1%29\"$ Combine like terms. Or factor out the common term $\"x-1\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "---------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"x%5E2-3x%2B2\"$ factors to $\"%28x-2%29%28x-1%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2-3x%2B2=%28x-2%29%28x-1%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28x-2%29%28x-1%29\"$ to get $\"x%5E2-3x%2B2\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );