Factor by grouping : 2a^2-5bc-2ab+5ac?

Question

please help

Answer 1

You can factor 2a from the 1st and 3rd terms, and 5c from the 2nd and 4th terms. To make things easier, first rearrange the terms

2a^2 - 2ab + 5ac - 5bc

2a(a - b) + 5c(a - b)

After this grouping, we can write

(2a + 5c)(a - b)

It is important to remember that you can change the order in which terms appear in an expression without changing its value.

Answer 2

Remember - factoring is all about finding repeating patterns in the equations and grouping terms that have the pattern and then factoring the repeated pattern out of the group (removing and multiply with what is remaining).

Note that a is repeated (practically) everywhere, so group all terms and factor out a^2:

a^2 (2 - 5 b c / a^2 - 2 b/a + 5 c/a)

Let b' = b / a and c' = c / a and replace (just to keep things simple).

a^2 (2 - 5 b' c' - 2 b' + 5 c')

Note that 2 and 5 c' form repeating patterns so group the terms that have these patterns and factor them out of each group.

a^2 ((2 - 2 b') + (5 c'- 5 b' c') ) ; grouping
a^2 ((2(1-b') + 5 c' (1-b')) ; factoring

OMG! Surprise! Now 1 - b' is a repeating pattern so group all terms and factor.

a^2 (1-b') (2 + 5 c')

Cool! Right? Now substitute back for b' and c':

a^2 (1 - b / a) (2 + 5 c / a) or

To replace a back into each factor we multiply each factor by a:

(a (1 - b / a)) (a(2 + 5 c / a))

ANS: (a - b) (2 a + 5 c)

Note that you may find other patterns and start with different groups but you should always end up in the same place. So don't be flustered on where to start. It might be somewhat more complicated - as this case - but it will give you insight into how to better approach the problem. Doing it all in your head can hurt the knogen.

Hope this helps.

Answer 3

Try rearranging the terms:

2a^2 + 5ac - 2ab - 5bc.

Factor out what each term has in common (the a and the -b).

a(2a +5c) - b(2a + 5c)

See if you can do the rest. (Hint: it uses the distributive property.)

Answer 4

2a^2 - 5bc - 2ab + 5ac
2a^2 + 5ac - 2ab - 5bc
(2a^2 + 5ac) + (-2ab - 5bc)
a(2a + 5c) - b(2a + c)
(a - b)(2a + 5c)

Answer 5

2a^2 + 5ac -2ab -5bc preparing

a(2a +5c) -b(2a+5b)

(a-b)(2a+5c)

Answer 6

2a^2 - 5bc-2ab+5ac
=2a^2 -2ab-5bc+5ac
=2a(a-b) - 5c(b-a)
= 2a(a-b)+5c(a-b)
=(a-b) (2a+5c)