The ExampleWe want to multiply 4×7. Let’s write it as (4)(7). Then,
(4)(7) = 28Now let’s replace 4 with its equivalent, 3+1. And let’s replace 7 with its equivalent, 5+2. Then,
(3+1)(5+2) = 28This seems to be a pretty strange way to write 4×7, doesn’t it? Yet in mathematics – in algebra-style notation – it is just as correct as 4×7. In this form, we can hopefully explain in an understandable way, how the algebra distributive property works.
Refer to the diagram to see how we can do each multiplication and addition in a way that is consistent and can help us to avoid confusion and mistakes.
(3)(5)+(3)(2)+(1)(5)+(1)(2) = (15) + (6) + (5) + (2) = 28Notice (perhaps surprisingly to the beginner) we arrive at the correct answer.
Generic Format of the Algebra Distributive PropertyA simple format for the algebra distributive property if it involves only four simple unknowns is,
(a+b)(c+d) = ac + ad + bc + bd [Answer]
Our Final ProblemNow, using what we’ve just learned, let’s try a new problem. We will combine constants and/or variables to make each of our unknowns. Let’s try,
(3x²+4y)(5x+2)Using the order of multiplication we’ve illustrated above, we get
(3x²+4y)(5x+2) = 3x²5x + 3x²2 + 4y5x + 4y2or,
(3x²+4y)(5x+2) = 15x³ + 6x² + 20yx + 8y [Answer]Do these really match? For confirmation, let’s pick a value for x and for y, say x = –1 and y =2. Then,
(3x²+4y)(5x+2) = (3(1)+4(2))(5(–1)+2) = –33and
15x³ + 6x² + 20yx + 8y = –15 + 6 –40 +16 = –33The results match, as well they should!
Note: You might also like Mathematical Powers – A Simple InsightReferences: ← Back to Math-Logic-Design