7ac^2+2bc^2-7ad^2-2bd^2

Question

Factoring completely 

Answer 1

Hello Zoe,

this looks very convoluted so let's make it more user-friendly.

1.By collecting all expressions that have the same variables (variables = letters) and the same number

7ac² and -7ad²

+2bc² and -2bd²

So the individual expressions factored will look like this:

7a(c² -d²)

+2b(c²-d²)

 

We are not done yet! Why? Because you can see that (c²-d²) is contained in both expressions:

So now we factor out (c²-d²) --> (7a+2b) (c²-d²)

And, (7a+2b) (c²-d²) is our final answer. You can use the "binomial chicken" to check your answer:

(7a+2b) (c²-d²)

= 7a*c² + 7a*(-d²) + 2b*c² + 2b(-d²)

= 7ac²-7ad²+2bc²-2bd²  

= 7ac²+2bc²-7ad²-2bd²

Generally, when you are trying to factor convoluted expressions, always try to find variables and numbers that can go together. Write them down in individual groups and then re-arrange them. Please be careful about the signs + and - will make a big difference if you leave them out or omit the altogether.

 

Answer 2

Hello Zoe,

7ac^2+2bc^2-7ad^2-2bd^2

This sum can be factored completely using the grouping method. We'll group first two terms and the next two terms.

(7ac^2+2bc^2)+(-7ad^2-2bd^2)

Next, we'll take the common out in first and second term using distributive property (ab + ac = a(b + c)).

So, c² is common in first two terms and -d² is common in next two terms

c²(7a + 2b) -d²(7a + 2b) (Note: Since, we took (-) sign out make sure there's a (+) sign inside the bracket because (-)(-) = +)

Now, if you see (7a + 2b) is common in both terms. So, take that out common

(7a + 2b)(c² - d²)

Next, use the formula (x² - y²) = ( x + y)(x - y)

(7a + 2b)(c + d)(c - d) ------------> answer

To cross check your answer multiply the factored expression using FOIL method and you'll get 7ac^2+2bc^2-7ad^2-2bd^2.

I hope this helps you.