Question

Find the product:
(i)$(x-3)(x+3)(x^{2} +9);$
(ii)$(2a+3)(2a-3)(4a^{2} +9);$
(iii) $(p+2)(p-2)(p^{2} +4);$
(iv)$(\frac{1}{2} m-\frac{1}{3} ) (\frac{1}{2} m+\frac{1}{3} ) (\frac{1}{4} m^{2} +\frac{1}{9} )$
(v) $(2x-y)(2x+y)(4x^{2} +y^{2} );$
(vi)$(2x - 3y)(2x + 3y)(4x^{2}+ 9y^{2} )$

Answer 1

$1. \: (x - 3)(x + 3)( {x}^{2} + 9) \\ \\ simplify \: by \:using \: (a-b)(a+b)={a}^2-{b}^2 \: \\ \\( {x}^{2} - 9)( {x}^{2}+9) \\ \\simplify\:again\:by\:using\:same\:identity \\ \\ {x}^4- 81$

$2. \: (2a - 3)(2a + 3)( {4a}^{2} + 9) \\ \\ simplify \: by \: using \: \\ (a - b)(a + b) = {a}^{2} - {b}^{2} \\ \\ (4 {a}^{2} - 9)(4 {a}^{2} + 9) \\ \\ use \: the \: same \: identity \: again \: and \: simplify \\ \\ {16a}^{2} - 81$

$3. \: (p + 2)(p - 2)( {p}^{2} + 4) \\ \\ simplify \: by \: using \\ (a + b)(a - b) = {a}^{2} - {b}^{2} \\ \\ ( {p}^{2} - 4)( {p}^{2} + 4) \\ \\ use \: the \: same \: identity \: again \: to \: simplify \\ \\ {p}^{4} - 16$

$4. \: (\frac{1}{2} m - \frac{1}{3} )( \frac{1}{2} m + \frac{1}{3} )( \frac{1}{4} {m}^{2} + \frac{1}{9} ) \\ \\ simplify \: by \: using \\ (a - b)(a + b) = {a}^{2} - {b}^{2} \\ \\ ( \frac{1}{4} {m}^{2} - \frac{1}{9} )( \frac{1}{4} {m}^{2} + \frac{1}{9} ) \\ \\ simplify \: by \: using \: the \: same \: identity \\ \\ \frac{1}{16} {m}^{4} - \frac{1}{81}$

$5. \: (2x - y)(2x + y)( {4x}^{2} + {y}^{2} ) \\ \\ simplify \: by \: using \\ (a - b)(a + b) = {a}^{2} - {b}^{2} \\ \\ ( {4x}^{2} - {y}^{2} )( {4x}^{2} + {y}^{2}) \\ \\ simplify \: by \: using \: the \: same \: identity \\ \\ ( {16x}^{4} - {y}^{4} )$

$6. \: (2x - 3y)(2x + 3y)( {4x}^{2} + 9 {y}^{2} ) \\ \\ simplify \: by \: using \\ (a - b)(a + b) = {a}^{2} - {b}^{2} \\ \\ ( {4x}^{2} - 9 {y}^{2} )( {4x}^{2} + {9y}^{2} ) \\ \\ simplify \: by \: using \: the \: same \: identity \\ \\ {16x}^{4} - 81 {y}^{4}$