Page:Elementary algebra (1896).djvu/190

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172 ALGEBRA.

product the sign is + or — according as the quantities composing it have like or unlike signs.

Note. The square terms are always positive.

The same laws hold whatever be the number of terms in the expression to be squared.

Rule. To find the square of any multinomial: to the sum of the squares of the several terms add twice the product (with the proper sign) of each term into each of the terms that follow it.

Ex. 1.	$\left(x-2y-3z\right)^{2}$	$=$	$x^{2}+4y^{2}+9z^{2}-2\cdot x\cdot 2y-2\cdot x\cdot 3z+2\cdot 2y\cdot 3z$
		$=$	$x^{2}+4y^{2}+9z^{2}-4xy-6xz+12yz$ .
Ex. 2.	$\left(1+2x-3x^{2}\right)^{2}$	$=$	$1+4x^{2}+9x^{4}+2\cdot 1\cdot 2x-2\cdot 3x^{2}-2\cdot 2x\cdot 3x^{2}$
		$=$	$1+4x^{2}+9x^{4}+4x-6x^{2}-12x^{3}$
		$=$	$1+4x-2x^{2}-12x^{3}+9x^{4}$ .

EXAMPLES XX. b.

Write the square of each of the following expressions:

1.	$a+3b$ .	3.	$x-5y$ .	5.	$3x-y$ .	7.	$9x-2y$ .
2.	$a-3b$ .	4.	$2x+3y$ .	6.	$3x+5y$ .	8.	$5ab-c$ .

9.	$a-h-c$ .	14.	$1^{.}xy+yz+zx$ .	18.	${\frac {a}{2}}-2b+{\frac {c}{4}}$ .
10.	$a+b-c$ .	15.	$x-y+a-b$ .	18.	${\frac {a}{3}}-3b-{\frac {3}{2}}$ .
12.	$2a-3b+4c$ .	16.	$2x+3y+a-2b$ .	19.	${\frac {a}{3}}-3b-{\frac {3}{2}}$ .
13.	$x^{2}-y^{2}-z^{2}$ .	17.	$m-n-p-q$ .	20.	${\frac {2}{3}}x^{2}-x+{\frac {3}{2}}$ .

TO CUBE A BINOMIAL.

191. By actual multiplication, we have

$\left(a+b\right)^{3}$	$=$	$\left(a+b\right)\left(a+b\right)\left(a+b\right)$
	$=$	$a^{3}+3a^{2}b+3ab^{2}+b^{3}$	. . . (1),
$\left(a-b\right)^{3}$	$=$	$a^{3}-3a^{2}b+3ab^{2}-b^{2}$	. . . (2).

From these results we obtain the following rule:—

Rule. To find the cube of any binomial: take the cube of the first term, three times the square of the first by the second, three times the first by the square of the second, and the cube of the last.