SOLUTION: How do you factor this special case polynomial? 4d squared + 36d + 81

Question 960518: How do you factor this special case polynomial?
4d squared + 36d + 81
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .

Now multiply the first coefficient by the last term to get .

Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?

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

Factors of :
1,2,3,4,6,9,12,18,27,36,54,81,108,162,324
-1,-2,-3,-4,-6,-9,-12,-18,-27,-36,-54,-81,-108,-162,-324

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

These factors pair up and multiply to .
1*324 = 324
2*162 = 324
3*108 = 324
4*81 = 324
6*54 = 324
9*36 = 324
12*27 = 324
18*18 = 324
(-1)*(-324) = 324
(-2)*(-162) = 324
(-3)*(-108) = 324
(-4)*(-81) = 324
(-6)*(-54) = 324
(-9)*(-36) = 324
(-12)*(-27) = 324
(-18)*(-18) = 324

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First Number	Second Number	Sum
1	324	1+324=325
2	162	2+162=164
3	108	3+108=111
4	81	4+81=85
6	54	6+54=60
9	36	9+36=45
12	27	12+27=39
18	18	18+18=36
-1	-324	-1+(-324)=-325
-2	-162	-2+(-162)=-164
-3	-108	-3+(-108)=-111
-4	-81	-4+(-81)=-85
-6	-54	-6+(-54)=-60
-9	-36	-9+(-36)=-45
-12	-27	-12+(-27)=-39
-18	-18	-18+(-18)=-36

From the table, we can see that the two numbers and add to (the middle coefficient).

So the two numbers and both multiply to and add to

Now replace the middle term with . Remember, and add to . So this shows us that .

Replace the second term with .

Group the terms into two pairs.

Factor out the GCF from the first group.

Factor out 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.

Combine like terms. Or factor out the common term

Condense the terms.

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

Answer:

So factors to .

In other words, .

Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).

------------------------------------------------------------------------------------------------------------------------

If you need more one-on-one help, email me at jim_thompson5910@hotmail.com. You can ask me a few more questions for free, but afterwards, I would charge you ($2 a problem to have steps shown or $1 a problem for answer only).

Alternatively, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Any amount is greatly appreciated as it helps me a lot. This donation is to support free tutoring. Thank you.

Jim
------------------------------------------------------------------------------------------------------------------------
RELATED QUESTIONS
4d squared + 16 Factor each Polynomial (answered by Alan3354,erika514)

how do I factor this polynomial? Thank you for your help :) 64x to the second... (answered by Fombitz)

Factor the Polynomial.... (answered by edjones)

x squared minus3y minus18 how do u factor this polynomial (answered by stanbon)

How do you factor and solve this polynomial fraction?:2x squared-28x+98 over x... (answered by elima)

-4r squared + 100s squared How do you factor this polynomial Is the answer... (answered by richwmiller)

how do you do you factor this polynomial 50g squared plus 40g plus 8. i just cant square... (answered by scott8148)

How do I factor this equation completely? 8k... (answered by haileytucki)

how do you simplify this equation... (answered by Fombitz)