document.write( "Question 102564: Can someone please, please help me with this one?\r
\n" ); document.write( "\n" ); document.write( "16x^4 - 40x^2 + 9 \n" ); document.write( "

Algebra.Com's Answer #74578 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
Let $\"t=x%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So we now get\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"16t%5E2+-+40t+%2B+9\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\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 $\"16t%5E2-40t%2B9\"$ , we can see that the first coefficient is $\"16\"$ , the second coefficient is $\"-40\"$ , and the last term is $\"9\"$ .

Now multiply the first coefficient $\"16\"$ by the last term $\"9\"$ to get $\"%2816%29%289%29=144\"$ .

Now the question is: what two whole numbers multiply to $\"144\"$ (the previous product) and add to the second coefficient $\"-40\"$ ?

To find these two numbers, we need to list all of the factors of $\"144\"$ (the previous product).

Factors of $\"144\"$ :

1,2,3,4,6,8,9,12,16,18,24,36,48,72,144

-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72,-144

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

These factors pair up and multiply to $\"144\"$ .

1*144 = 144
2*72 = 144
3*48 = 144
4*36 = 144
6*24 = 144
8*18 = 144
9*16 = 144
12*12 = 144
(-1)*(-144) = 144
(-2)*(-72) = 144
(-3)*(-48) = 144
(-4)*(-36) = 144
(-6)*(-24) = 144
(-8)*(-18) = 144
(-9)*(-16) = 144
(-12)*(-12) = 144

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-40\"$ :

\n" ); document.write( "

First Number	Second Number	Sum
1	144	1+144=145
2	72	2+72=74
3	48	3+48=51
4	36	4+36=40
6	24	6+24=30
8	18	8+18=26
9	16	9+16=25
12	12	12+12=24
-1	-144	-1+(-144)=-145
-2	-72	-2+(-72)=-74
-3	-48	-3+(-48)=-51
-4	-36	-4+(-36)=-40
-6	-24	-6+(-24)=-30
-8	-18	-8+(-18)=-26
-9	-16	-9+(-16)=-25
-12	-12	-12+(-12)=-24

From the table, we can see that the two numbers $\"-4\"$ and $\"-36\"$ add to $\"-40\"$ (the middle coefficient).

So the two numbers $\"-4\"$ and $\"-36\"$ both multiply to $\"144\"$ and add to $\"-40\"$

Now replace the middle term $\"-40t\"$ with $\"-4t-36t\"$ . Remember, $\"-4\"$ and $\"-36\"$ add to $\"-40\"$ . So this shows us that $\"-4t-36t=-40t\"$ .

$\"16t%5E2%2Bhighlight%28-4t-36t%29%2B9\"$ Replace the second term $\"-40t\"$ with $\"-4t-36t\"$ .

$\"%2816t%5E2-4t%29%2B%28-36t%2B9%29\"$ Group the terms into two pairs.

$\"4t%284t-1%29%2B%28-36t%2B9%29\"$ Factor out the GCF $\"4t\"$ from the first group.

$\"4t%284t-1%29-9%284t-1%29\"$ Factor out $\"9\"$ 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.

$\"%284t-9%29%284t-1%29\"$ Combine like terms. Or factor out the common term $\"4t-1\"$

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

Answer:

So $\"16%2At%5E2-40%2At%2B9\"$ factors to $\"%284t-9%29%284t-1%29\"$ .

In other words, $\"16%2At%5E2-40%2At%2B9=%284t-9%29%284t-1%29\"$ .

Note: you can check the answer by expanding $\"%284t-9%29%284t-1%29\"$ to get $\"16%2At%5E2-40%2At%2B9\"$ or by graphing the original expression and the answer (the two graphs should be identical).

\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Since $\"t=x%5E2\"$ , we can replace each t with $\"x%5E2\"$ to get \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%284x%5E2-9%29%284x%5E2-1%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );