Simplifying Expressions like $-a(3b - 2c - d)$

Now we're ready to look at several extensions of the distributive law. Recall that the ‘basic model’ of the distributive law is: For all real numbers $\,a\,,$ $\,b\,,$ and $\,c\,,$ $\,a(b+c) = ab + ac\,.$

There may be more than two terms in the parentheses:

$a(b + c + d) = ab + ac + ad$

$a(b + c + d + e) = ab + ac + ad + ae$

and so on. All the usual rules for dealing with signed terms hold. For example:

$-a(2b + c + 4d + f) = -2ab - ac - 4ad - af$

Remember to determine the sign (plus or minus) first, the numerical part next, and the variable part last.

Example

Question: Simplify: $\,a(b - c + e)$

Answer: $ab - ac + ae$
Do not change the order of the letters: write $\,ab-ac+ae\,,$  not (say)  $\,ba-ac+ea\,.$ Even though answers like ‘$\,ba-ac+ea\,$’ are correct, they are not recognized as correct by this program.

Practice