Page:Elementary algebra (1896).djvu/191

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INVOLUTION. 173

If the binomial be the sum of the quantities, all signs will be +; if the difference of two quantities, the signs will be alternately + and —, commencing with the first.

EXAMPLES XX. c.

Write the cube of each of the following expressions:

1.	$x+a$ .	5.	$x2+4y^{2}$ .	9.	$a-{\frac {2b}{3}}$ .	11.	${\frac {x^{2}}{3}}-3x$ .
2.	$x-a$ .	6.	$4x^{2}-5y^{2}$ .
3.	$x-2y$ .	7.	$2a^{3}-3b^{2}$ .	10.	${\frac {a}{3}}+2$ .	12.	${\frac {a}{6}}+2x$ .
4.	$2ab-3c$ .	8.	$5x^{5}-4y^{4}$ .

TO CUBE ANY MULTINOMIAL.

192. Consider a trinomial:

$\left(a+b+c\right)$	$=$	$\left[a+\left(b+c\right)\right]^{3}$
⁠	$=$	$a^{3}+3a^{2}\left(b+c\right)+3a\left(b+c\right)^{2}+\left(b+c\right)^{3}$
	$=$	$a^{3}+b^{3}+c^{3}+3a^{2}\left(b+c\right)+3b^{2}\left(a+c\right)$
	$+$	$3c^{2}\left(a+b\right)+6abc$ .

Rule. To cube a multinomial: take the cube of each term, three times the square of each term by every other term, and six times the product of every three different terms. The signs are determined by the law of signs for multiplication.

EXAMPLES XX. d.

Write the cube of each of the following expressions

1.	$1+x+x^{2}$ .	4.	$a+bx+x^{2}$ .	7.	${\frac {x}{2}}+{\frac {x^{2}}{3}}-x^{3}$ .
2.	$1+x-x^{2}$ .	5.	$2a+bx-cx^{2}$ .	8.	$1+x+x^{2}+x^{3}$ .
3.	$1-2x+3x^{2}$ .	5.	$3x+2x^{2}-x^{3}$ .	8.	$2-3x+x^{2}+2x^{3}$ .

TO RAISE A BINOMIAL TO ANY POWER EXPRESSED BY A POSITIVE INTEGER [BINOMIAL THEOREM].

193. By actual multiplication, we obtain the following identities:

$\left(a+b\right)^{3}$	$=$	$a^{3}+3a^{2}b+3ab^{2}+b$ ;
$\left(a+b\right)^{4}$	$=$	$a^{4}+4a^{3}b+6a^{2}b^{2}+4ab^{3}+b^{4}$