If the pupil does not perceive the force of this reasoning, it
will be best to repeat the argument with numbers, thus : Let it
be proposed to multiply 8—5 by 6—2; that is, the quantity
8— 5 is to be repeated as many times as there are units in
6—2. If we multiply 8—5 by 6, we obtain 48—30; that is,
we have repeated 8— 5 six times. But it was only required to
repeat the multiplicand ^wr times, or 6—2. We must there-
fore diminish this product by twice 8 — 5, which is 16—10;
and this subtraction is performed by changing the signs of the
subtrahend. Hence we have

48-80-16+10,

which is equal to 12. This result is obviously correct; for
8— 5 is equal to 3, awd 6— 2 is equal to 4; that is, it was re-
quired to multiply 3 by 4, the result of which is 12, as found
above.
We have thus obtained the following results:

+ax{+b)=+ab,
+ax(— 6)=— a6,
— ax(+6)=--a6,
— ax(— 6)=+ai,

from which we perceive that when the two factors have like
signSy Hie prodvjct is positive; and when the two factors have un-
like signs, the product is negative.
61. Hence, for the multiplication of polynomials, we have
the following general

RULE.
Multiply each term of -the muUiplinand by each term of the rr^ul-



36 ALGEBRA.
tipliery and add together all the partial products^ observing thai
like signs require + in the prodrug and unlike signs —.
EXAJCFLBa
1. 2.
