Page:Elementary algebra (1896).djvu/101

${\begin{aligned}\mathrm {Ex.} \ \mathbf {2.} \qquad \quad 64a^{3}+1&=(4a)^{3}+(1)^{3}\\&=(4a+1)(16a^{2}-4a+1).\\\end{aligned}}$

We may usually omit the intermediate step and write the factors at once.

$\mathrm {Ex.} \ \mathbf {3.} \quad 343a^{6}-27x^{3}=(7a^{2}-3x)(49a^{4}+21a^{2}x+9x^{2}).$
$\mathrm {Ex.} \ \mathbf {4.} \quad 8x^{9}+729=(2x^{3}+9)(4x^{6}-18x^{3}+81).$

EXAMPLES X. m.

Resolve into factors:

$\mathbf {1.} \ \ x^{3}-y^{3}.$
$\mathbf {2.} \ \ x^{3}+y^{3}.$
$\mathbf {3.} \ \ x^{3}-1.$
$\mathbf {4.} \ \ 1+a^{3}.$

$\mathbf {5.} \ \ 8x^{3}-y^{3}.$
$\mathbf {6.} \ \ x^{3}+8y^{3}.$
$\mathbf {7.} \ \ 27x^{3}+1.$
$\mathbf {8.} \ \ 1-8y^{3}.$

$\mathbf {9.} \ \ a^{3}b^{3}-c^{3}.$
$\mathbf {10.} \ \ 8x^{3}+27y^{3}.$
$\mathbf {11.} \ \ 1-343x^{3}.$
$\mathbf {12.} \ \ 64+y^{3}.$

$\mathbf {13.} \ \ 125+a^{3}.$
$\mathbf {14.} \ \ 216-a^{3}.$
$\mathbf {15.} \ \ a^{3}b^{3}+512.$
$\mathbf {16.} \ \ 1000y^{3}-1.$

$\mathbf {17.} \ \ x^{3}+64y^{3}.$
$\mathbf {18.} \ \ 27-1000x^{3}.$
$\mathbf {19.} \ \ a^{3}b^{3}+216c^{3}.$
$\mathbf {20.} \ \ 343-8x^{3}.$
$\mathbf {21.} \ \ a^{3}+27b^{3}.$
$\mathbf {22.} \ \ 27x^{3}-64y^{3}.$
$\mathbf {23.} \ \ 125x^{3}-1.$
$\mathbf {24.} \ \ 216p^{3}-343.$

$\mathbf {25.} \ \ x^{3}y^{3}+z^{3}.$
$\mathbf {26.} \ \ a^{3}b^{3}c^{3}-1.$
$\mathbf {27.} \ \ 343x^{3}+1000y^{3}.$
$\mathbf {28.} \ \ 729a^{3}-64b^{3}.$
$\mathbf {29.} \ \ 8a^{3}b^{3}+125x^{3}.$
$\mathbf {30.} \ \ x^{3}y^{3}-216z^{3}.$
$\mathbf {31.} \ \ x^{6}-27y^{3}.$
$\mathbf {32.} \ \ 64x^{6}+125y^{3}.$

$\mathbf {33.} \ \ 8x^{3}-z^{6}.$
$\mathbf {34.} \ \ 216x^{6}-b^{3}.$
$\mathbf {35.} \ \ a^{3}+343b^{3}.$
$\mathbf {36.} \ \ a^{6}+729b^{3}.$
$\mathbf {37.} \ \ 8x^{3}-729y^{6}.$
$\mathbf {38.} \ \ p^{3}q^{3}-27x^{3}.$
$\mathbf {39.} \ \ z^{3}-64y^{6}.$
$\mathbf {40.} \ \ x^{3}y^{3}-512.$

103. Miscellaneous Cases of Resolution into Factors.

$\mathrm {Ex.} \ \mathbf {1.} \quad$ Resolve into factors $16a^{4}-81b^{4}.$

${\begin{aligned}16a^{4}-81b^{4}&=(4a^{2}+9b^{2})(4a^{2}-9b^{2})\\&=(4a^{2}+9b^{2})(2a+3b)(2a-3b).\\\end{aligned}}$

$\mathrm {Ex.} \ \mathbf {2.} \quad$ Resolve into factors $x^{6}-y^{6}.$

${\begin{aligned}x^{6}-y^{6}&=(x^{3}+y^{3})(x^{3}-y^{3})\\&=(x+y)(x^{2}-xy+y^{2})(x-y)(x^{2}+xy+y^{2}).\\\end{aligned}}$

Note. When an expression can be arranged either as the difference of two squares, or as the difference of two cubes, each of the methods explained in Arts. 98, 102 will be applicable. It will, however, be found simplest to first use the rule for resolving into factors the difference of two squares.

In all cases where an expression to be resolved contains a simple factor common to each of its terms, this should be first taken outside a bracket as explained in Art. 90.