## Exercises - Factoring

1. Factor completely:

1. $\displaystyle{3a^2 - 8a + 4}$

${\displaystyle{(3a-2)(a-2)}}$

2. $\displaystyle{16a^4 - 1}$

${ \begin{array}{rcl} &=& (4a^2 - 1)(4a^2 + 1)\\ &=& \fbox{$(2a-1)(2a+1)(4a^2 +1)$} \end{array}}$

3. $\displaystyle{2x^2 + 2xy - 3x - 3y}$

${ \begin{array}{rcl} &=& (2x^2 + 2xy) + (-3x - 3y)\\ &=& 2x(x + y) - 3(x + y) \\ &=& \fbox{$(2x - 3)(x + y)$} \end{array}}$

4. $\displaystyle{4z^3 - 32}$

${ \begin{array}{rcl} &=& 4(z^3-8)\\ &=& \fbox{$4(z-2)(z^2+2z+4)$} \end{array}}$

5. $\displaystyle{x^2 - 2x + 1 - z^2}$

${ \begin{array}{rcl} &=& (x^2 - 2x + 1) - z^2\\ &=& (x - 1)^2 - z^2\\ &=& \fbox{$(x - 1 + z)(x - 1 - z)$} \end{array}}$

6. $\displaystyle{x^3 + 2x^2 - 3x - 6}$

${ \begin{array}{rcl} &=& (x^3+2x^2) + (-3x-6)\\ &=& x^2(x+2) - 3(x+2)\\ &=& \fbox{$(x^2 - 3)(x+2)$} \end{array}}$

7. $\displaystyle{24x + 144 + x^2}$

${\displaystyle{(x+12)^2}}$

8. $\displaystyle{8x^3 - 1}$

${\displaystyle{(2x-1)(4x^2+2x+1)}}$

9. $\displaystyle{6x^3 + 48}$

${ \begin{array}{rcl} &=& 6(x^3+8)\\ &=& \fbox{$6(x+2)(x^2-2x+4)$} \end{array}}$

10. $\displaystyle{9x^2 - 30x + 25}$

${\displaystyle{(3x-5)^2}}$

11. $\displaystyle{9x^2 + 6xy + y^2 - 4}$

${ \begin{array}{rcl} &=& (9x^2 + 6xy + y^2) - 4\\ &=& (3x+y)^2 - 4\\ &=& \fbox{$(3x+y-2)(3x+y+2)$} \end{array}}$

12. $\displaystyle{x^4 + xy^3 + 4yx^3 + 4y^4}$

${ \begin{array}{rcl} &=& (x^4 + xy^3) + (4yx^3 + 4y^4)\\ &=& x(x^3+y^3) + 4y(x^3+y^3)\\ &=& (x+4y)(x^3+y^3)\\ &=& \fbox{$(x+4y)(x+y)(x^2-xy+y^2)$} \end{array}}$

13. $\displaystyle{2x^4 - 2y^2 - 4y - 2}$

${ \begin{array}{rcl} &=& 2(x^4-y^2-2y-1)\\ &=& 2(x^4 - (y^2+2y+1))\\ &=& 2(x^4 - (y+1)^2)\\ &=& 2(x^2 + (y+1))(x^2 - (y+1))\\ &=& \fbox{$2(x^2+y+1)(x^2-y-1)$} \end{array}}$

14. $\displaystyle{2x^3+4x^2-x-2}$

${ \begin{array}{rcl} &=& (2x^3+4x^2) + (-x-2)\\ &=& 2x^2(x+2) - (x+2)\\ &=& \fbox{$(2x^2-1)(x+2)$} \end{array}}$

15. $\displaystyle{16b^3-2a^3}$

${ \begin{array}{rcl} &=& 2(8b^3-a^3)\\ &=& \fbox{$2(2b - a)(4b^2 +2ab + a^2)$} \end{array}}$

16. $\displaystyle{2x^4 - 6x^3 - 20x^2}$

${ \begin{array}{rcl} &=& 2x^2(x^2-3x-10)\\ &=& \fbox{$2x^2(x-5)(x+2)$} \end{array}}$

17. $\displaystyle{128x^6 - 2y^6z^{12}}$

${ \begin{array}{rcl} &=& 2(64x^6-y^6 z^12)\\ &=& 2(8x^3-y^3 z^6)(8x^3+y^3 z^6)\\ &=& \fbox{$2(2x-yz^2)(2x+yz^2)(4x^2+2xyz^2+y^2z^4)(4x^2-2xyz^2+y^2z^4)$} \end{array}}$

18. $\displaystyle{x^3 + y^3 + 3y^2 + 3y + 1}$

${ \begin{array}{rcl} &=& x^3 + (y^3 + 3y^2 + 3y + 1)\\ &=& x^3 + (y+1)^3\\ &=& (x + (y+1))(x^2 - x(y+1) + (y+1)^2)\\ &=& \fbox{$(x+y+1)(x^2-xy-x+y^2+2y+1)$} \end{array}}$

19. $\displaystyle{9x^2 - 6xy + y^2 - 25}$

${ \begin{array}{rcl} &=& (9x^2 - 6xy + y^2) - 25\\ &=& (3x-y)^2 - 25\\ &=& \fbox{$(3x-y-5)(3x-y+5)$} \end{array}}$

2. Factor completely:

1. $\displaystyle{x^3-x}$

${ \begin{array}{rcl} &=& x(x^2 - 1)\\ &=& \fbox{$x(x-1)(x+1)$} \end{array}}$

2. $\displaystyle{27x^3-1}$

${\displaystyle{(3x-1)(9x^2+3x+1)}}$

3. $\displaystyle{4x^4-13x^2+9}$

${ \begin{array}{rcl} &=& (4x^2-9)(x^2-1)\\ &=& \fbox{$(2x+3)(2x-3)(x+1)(x-1)$} \end{array}}$

4. $\displaystyle{3x^2-17x+10}$

${\displaystyle{(x-5)(3x-2)}}$

5. $\displaystyle{3x^3-3xy-2x^2y+2y^2}$

${ \begin{array}{rcl} &=& (3x^3-3xy)+(-2x^2y+2y^2)\\ &=& 3x(x^2-y)-2y(x^2-y)\\ &=& \fbox{$(3x-2y)(x^2-y)$} \end{array}}$

6. $\displaystyle{6x^2+7x-3}$

${\displaystyle{(2x+3)(3x-1)}}$

7. $\displaystyle{8y^6+125y^3}$

${ \begin{array}{rcl} &=& y^3(8y^3+125y^3)\\ &=& \fbox{$y^3(2y+5)(4y^2-10y+25)$} \end{array}}$

8. $\displaystyle{9x^2+6x+1}$

${\displaystyle{(3x+1)^2}}$

9. $\displaystyle{3x^2+18x+15}$

${ \begin{array}{rcl} &=& 3(x^2+6x+5)\\ &=& \fbox{$3(x+5)(x+1)$} \end{array}}$

10. $\displaystyle{4ax-2bx+2ay-by}$

${ \begin{array}{rcl} &=& (4ax-2bx) + (2ay-by)\\ &=& 2x(2a-b)+y(2a-b)\\ &=& \fbox{$(2x+y)(2a-b)$} \end{array}}$

11. $\displaystyle{32a^4c-2b^4c}$

${ \begin{array}{rcl} &=& 2c(16a^4-b^4)\\ &=& 2c(4a^2-b^2)(4a^2+b^2)\\ &=& \fbox{$2c(2a-b)(2a+b)(4a^2+b^2)$} \end{array}}$

12. $\displaystyle{a^6-64}$

${ \begin{array}{rcl} &=& (a^3-8)(a^3+8)\\ &=& \fbox{$(a-2)(a^2+2a+4)(a+2)(a^2-2a+4)$} \end{array}}$

13. $\displaystyle{2x^4-17x^2-9}$

${ \begin{array}{rcl} &=& (2x^2+1)(x^2-9)\\ &=& \fbox{$(2x^2+1)(x+3)(x-3)$} \end{array}}$

14. $\displaystyle{2x^4-6x^3-20x^2}$

${ \begin{array}{rcl} &=& 2x^2(x^2-3x-10)\\ &=& \fbox{$2x^2(x-5)(x+2)$} \end{array}}$

15. $\displaystyle{3x^4+375x}$

${ \begin{array}{rcl} &=& 3x(x^3+125)\\ &=& \fbox{$3x(x+5)(x^2-5x+25)$} \end{array}}$

16. $\displaystyle{xy^2+y^2-4x-4}$

${ \begin{array}{rcl} &=& (xy^2+y^2) + (-4x-4)\\ &=& y^2(x+1) - 4(x+1)\\ &=& (y^2-4)(x+1)\\ &=& \fbox{$(y-2)(y+2)(x+1)$} \end{array}}$