factor polynomial completely

Question

Factor Polynomial completely

 

1) 27y²-57y+28

 

2)16m²-121

Answer 1

The 2nd one is easier than the first so I'll start with that

(x + y)(x - y) = x2 - y2 is the formula. When you multiply the two parenthetical terms (x+y) and (x-y) together, you get
x2 + xy - xy - y2. Since the middle two terms cancel each other out, the result is x2 - y2 which is the same format as your equation.

So in your problem you simply need to figure out the square root of each of your terms. The square root of 16m2 is 4m (4m x 4m = 16m2). The square root of your 2nd term is 11. So going with the known formula above, you would end up with (4m + 11)(4m - 11) as your answer. Multiple those two together to confirm that your math was correct to see that the result is 16m2 + 44m - 44m - 121 = 16m2 - 121.

Now the first equation. You are looking for some form of (ax + by)(cx + dy), with a,b,c, and d being your coefficients (numbers). it takes some trial and error to find the answer. First, you need to figure out what possible coefficients you can have:
for your first term (the term containing the y² you can factor the number as 1x27, 3x9, 9x3, or 27x1. For your final term, you can have 1x28, 2x14, 4x7, 7x4, 14x2, or 28x1.

 

Explaining how to reason through the process of selecting your coefficients takes a bit more time than works in this venue.  I can give you the answer here (3y - 4)(9y - 7), but it won't help you know how to do it yourself on future problems.  Feel free to contact me for tutoring help if desired.

Answer 2

1) 27y²-57y+28, factor completely.

 

-multiply first and last coefficients:  27*28=756

 

-find 2 factors of 756 that add to get -57, the middle term =(-21)+(-36)

 

-substitute the two new factors for the middle term to get: 1) 27y²-21y-36y+28

 

-separate into 2 bracketed expressions:  (27y²-21y)(-36y+28)

 

-factor out the largest possible coefficient and variable from each bracket:3y(9y-7)-4(9y-7)

 

-combine the external terms into one bracket using the Distributive Property:(3y-4)(9y-7)

 

 

Answer 3

For the first one,

 

You will have to factor using FOIL.  The product of the first terms in the binomial must be 27y², but the product of the last terms in the binomial must be 28.  The sum of the products in the outer terms and inner terms must be -57y.  Therefore, your factor will be in the form.

 

(ay - b)(cy - d)

 

where  a, b, c, and d are integers.

 

We perform a guess and check.

 

(9y - 7)(3y - 4)

 

 

This satisfies the 1st and last term in the trinomial, but does it satisfy the sum?   Lets check.

 

Product of outer + product of inner = middle term

 

(9y * -4) + (-7 * 3y) = -36y - 21y = -57y 

 

 

As you can see, it does.   So your factor is

 

(9y - 7)(3y - 4)