How do you factor the expression #63cd^2 − 234cd + 135c#?

Question

Avery Alexander · Accepted Answer

#63cd^2-234cd+135c=9c(7d-5)(d-3)#

All of the terms are divisible by #9c#, so separate that out as a factor first: #63cd^2-234cd+135c=9c(7d^2-26d+15)# Factor the remaining quadratic expression using an AC method: Look for a pair of factors of #AC=7*15=105# with sum #B=26# The pair #21, 5# works. Use that to split the middle term and factor by grouping: #7d^2-26d+15# #=7d^2-21d-5d+15# #=(7d^2-21d)-(5d-15)# #=7d(d-3)-5(d-3)# #=(7d-5)(d-3)# Putting it together: #63cd^2-234cd+135c=9c(7d-5)(d-3)#

Nathan Axel · Answer

undefined To factor the expression 63cd^2 − 234cd + 135c, you can follow these steps:

1. Find the greatest common factor (GCF) of all the terms. In this case, the GCF is 9c.
2. Divide each term by the GCF: 
   - $ \frac{63cd^2}{9c} = 7d^2 $
   - $ \frac{-234cd}{9c} = -26d $
   - $ \frac{135c}{9c} = 15 $
3. Rewrite the expression using the factored GCF: 
   $ 9c(7d^2 - 26d + 15) $
4. Factor the quadratic expression inside the parentheses: 
   - Identify two numbers that multiply to give the constant term (15) and add to give the coefficient of the middle term (-26). These numbers are -5 and -3.
   - Rewrite the middle term using the two numbers: 
     $ 7d^2 - 5d - 21d + 15 $
5. Group the terms and factor by grouping: 
   $ 7d(d - 5) - 3(d - 5) $
6. Factor out the common binomial factor: 
   $ (7d - 3)(d - 5) $
7. Combine the factored GCF with the factored quadratic expression: 
   $ 9c(7d - 3)(d - 5) $

So, the factored expression is $ 9c(7d - 3)(d - 5) $.