I don't get these Algebra concepts?

Question

Factor by grouping:

Factor the GCF out of 2 terms at a time in order to get a common binomial. Complete the factoring by removing the common binomial and writing a product of two binomials.

What's a binomial, I don't understand how to do it with these words???

it's on my homework. Here's the equations:

4x^2-14x+14x-49

and

18x^4y^2-24^3y^3+12x^2y^3

And also, what is the zero product property???

how do I do it???

2x^2+30x+108=0

I have so much more questions!!!! =(

it's too hard.........

Answer 1

Binomial means an expression containing two terms.(like 2x+3y)

You have to factor by taking out a common binomial term.

4x^2-14x+14x-49

from the first two term take out 2x as common and from the last two term take out 7 as common.

= 2x(2x-7) + 7(2x-7)

removing the common binomial(2x-7)

= (2x-7)(2x+7)

Answer 2

First one 4x^2-14x+14x-49

Notice the +14x and -14x cancel and you are left with 4x^2-49

  

4x^2-49 = (2x+?)(2x-?)

            = (2x+7)(2x-7)  difference between two square roots

classic factoring problem to remember: difference between squares.

Next problem: 18x^4y^2-24^3y^3+12x^2y^3

I think something is wrong...24^3 seems out of place

however if not pull out a 6x^2y^2

from the terms 18x^4y^2 and 12x^2y^3  to get

6x^2y^2(3x^2+2y)+ 24^3y^3

Now you can pull another y^2 out to get

y^2{6x^2(3x^2+2y)+24^3y}

Last 2x^2+30x+108=0

Your just putting everything on the left side and setting to 0.

But that's not a zero product property. A zero product property I would think is that when you multiply by 0 the answer is 0.

Dividing everything by 2 gives

x^2+15x+54 which factors to (x+?)(x+?)

= (x+9)(x+6) = 0

This gives two equations and two solutions:

x+9 = 0   and  x+6=0

Solutions are x=-9 or x=-6

done

Answer 3

4x^2 - 14x + 14x - 49

The GCF of 4x^2 and -14x is 2x.

The GCF of 14x and - 49 is 7.

2x(2x - 7) + 7(2x - 7) = (2x + 7)(2x - 7)

Note you could have just noticed that -14x + 14x = 0

in which case you would have 4x^2 - 49 which could

be rewritten as the difference of two squares:

(2x)^2 - (7)^2 = (2x - 7)(2x + 7)

18x^4y^2 - 24x^3y^3 + 12x^2y^3 = 6x^2y^2(3x^2 - 4xy + 2y)

The zero product property states that anything multiplied by 0 is 0.

What do you want me to do with the 2x^2 + 30x + 108 = 0?

Find roots?

2(x^2 + 15x + 54) = 0

2(x + 9)(x + 6) = 0

x = - 9, x = -6.

I assume you want to use the

idea of the zero product here.

Notice that when x = -9, you have

2(-9 + 9)(-9 + 6) = 0

2(0)(-3) = 0

0 = 0

When anything gets multiplied by zero you get 0.

I hope I helped you a bit.