How do you factor by grouping #5n^2 + 19n + 12#?

Question

Savannah Anderson · Accepted Answer

#=> (5n+4)(n+3)#

Factor by grouping? This is just normal factoring, no grouping is needed:

#5n^2+19n+12 = (5n+4)(n+3)#

If you want to use factoring by grouping anyway, then you have to break up the middle term with #n# into pieces that sum to #19n# in total.

#=(5n^2+15n) + (4n + 12)#

#=5n(n+3) + 4(n+3)#

#=(5n+4)(n+3)#

Eva Abramson · Answer

#(5n+4)(n+3)#

When we do factoring by grouping, we want to split the #b# term (coefficient on #n#) into two terms so we can factor two separate expression.

#5n^2+color(blue)(19n)+12# can be rewritten as

#5n^2+color(blue)(15n+4n)+12#

Notice, the blue terms are equivalent, so I didn't change the meaning of this expression.

Now, we can factor!

#color(red)(5n^2+15n)+color(lime)(4n+12)#

We can factor a #5n# out of the red expression, and a #4# out of the green expression. We get

#color(red)(5n(n+3))+color(lime)(4(n+3))#

which can be rewritten as

#(5n+4)(n+3)#

Hope this helps!

Eva Abramson · Answer

Use an AC method to split the middle term, then factor by grouping to find:

#5n^2+19n+12 = (5n+4)(n+3)#

Given: #5n^2+19n+12# Use an AC Method to find the split we need. Look for a pair of factors of #AC = 5 * 12 = 60# with sum #B=19#. The pair #15, 4# works in that #15 * 4 = 60# and #15+4=19#. Using this pair to split the middle term, we can then factor by grouping: #5n^2+19n+12 = (5n^2+15n)+(4n+12)# #color(white)(5n^2+19n+12) = 5n(n+3)+4(n+3)# #color(white)(5n^2+19n+12) = (5n+4)(n+3)#

Savannah Anderson · Answer

undefined To factor by grouping, first, multiply the leading coefficient (5) by the constant term (12) to get 60. Then find two numbers that multiply to 60 and add to the middle coefficient (19). These numbers are 15 and 4. Rewrite the middle term using these numbers. So, the expression becomes 5n^2 + 15n + 4n + 12. Group the first two terms and the last two terms together. Factor out the greatest common factor from each pair. In this case, factor out 5n from the first pair and 4 from the second pair. This leaves you with 5n(n + 3) + 4(n + 3). Factor out the common binomial factor (n + 3). So, the factored expression is (5n + 4)(n + 3).