Answer:
a. [tex](x - 3)^2 + 16[/tex]
b. [tex]8(x -7)^2[/tex]
c. [tex](a^2 - 1)(7x - 6)[/tex] or [tex](a+1)(a-1)(7x-6)[/tex]
d. [tex](x^2-4)(x^2+3)[/tex] or [tex](x-2)(x+2)(x^2+3)[/tex]
e. [tex](a^n+b^n)(a^n-b^n)(a^{2n} +b^{2n})[/tex]
Step-by-step explanation:
[tex]a.\ (x + 1)^2 - 8(x - 1) + 16[/tex]
Expand
[tex](x + 1)(x + 1) - 8(x - 1) + 16[/tex]
Open brackets
[tex]x^2 + x + x + 1 - 8x + 8 + 16[/tex]
[tex]x^2 + 2x + 1 - 8x + 24[/tex]
Collect Like Terms
[tex]x^2 + 2x - 8x+ 1 + 24[/tex]
[tex]x^2 - 6x+ 25[/tex]
Express 25 as 9 + 16
[tex]x^2 - 6x+ 9 + 16[/tex]
Factorize:
[tex]x^2 - 3x - 3x + 9 + 16[/tex]
[tex]x(x -3)-3(x - 3) + 16[/tex]
[tex](x - 3)(x - 3) + 16[/tex]
[tex](x - 3)^2 + 16[/tex]
[tex]b.\ 8(x - 3)^2 - 64(x-3) + 128[/tex]
Expand
[tex]8(x - 3)(x - 3) - 64(x-3) + 128[/tex]
[tex]8(x^2 - 6x+ 9) - 64(x-3) + 128[/tex]
Open Brackets
[tex]8x^2 - 48x+ 72 - 64x+192 + 128[/tex]
Collect Like Terms
[tex]8x^2 - 48x - 64x+192 + 128+ 72[/tex]
[tex]8x^2 -112x+392[/tex]
Factorize
[tex]8(x^2 -14x+49)[/tex]
Expand the expression in bracket
[tex]8(x^2 -7x-7x+49)[/tex]
Factorize:
[tex]8(x(x -7)-7(x-7))[/tex]
[tex]8((x -7)(x-7))[/tex]
[tex]8(x -7)^2[/tex]
[tex]c.\ 7a^2x - 6a^2 - 7x + 6[/tex]
Factorize
[tex]a^2(7x - 6) -1( 7x - 6)[/tex]
[tex](a^2 - 1)(7x - 6)[/tex]
The answer can be in this form of further expanded as follows:
[tex](a^2 - 1^2)(7x - 6)[/tex]
Apply difference of two squares
[tex](a+1)(a-1)(7x-6)[/tex]
[tex]d.\ x^4 - x^2 - 12[/tex]
Express [tex]x^4[/tex] as [tex]x^2[/tex]
[tex](x^2)^2 - x^2 - 12[/tex]
Expand
[tex](x^2)^2 +3x^2- 4x^2 - 12[/tex]
[tex]x^2(x^2+3) -4(x^2+3)[/tex]
[tex](x^2-4)(x^2+3)[/tex]
The answer can be in this form of further expanded as follows:
[tex](x^2-2^2)(x^2+3)[/tex]
Apply difference of two squares
[tex](x-2)(x+2)(x^2+3)[/tex]
[tex]e.\ a^{4n} -b^{4n}[/tex]
Represent as squares
[tex](a^{2n})^2 -(b^{2n})^2[/tex]
Apply difference of two squares
[tex](a^{2n} -b^{2n})(a^{2n} +b^{2n})[/tex]
Represent as squares
[tex]((a^{n})^2 -(b^{n})^2)(a^{2n} +b^{2n})[/tex]
Apply difference of two squares
[tex](a^n+b^n)(a^n-b^n)(a^{2n} +b^{2n})[/tex]
