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With so many different tools used to factor, it is prudent to have a section to determine the best strategy to factor.

Factoring Hints

1. Look for any factor to simplify the polynomial before you start!
2. If you have two terms, look for a sum or difference of squares or cubes.

\(a^2 – b^2  =  (a + b)(a – b)\)\(a^3 – b^3 = (a – b)(a^2 + ab + b^2)\)\(a^3 + b^3 = (a + b)(a^2 – ab + b2)\)

1. If you have three terms, see if the master product method works.
2. If you have four terms, see if factoring by grouping works.

Questions

Factor each completely.

1. \(24ac-18ab+60dc-45db\)
2. \(2x^2-11x+15\)
3. \(5u^2-9uv+4v^2\)
4. \(16x^2+48xy+36y^2\)
5. \(-2x^3+128y^3\)
6. \(20uv-60u^3-5xv+15xu^2\)
7. \(54u^3-16\)
8. \(54-128x^3\)
9. \(n^2-n\)
10. \(5x^2-22x-15\)
11. \(x^2-4xy+3y^2\)
12. \(45u^2-150uv+125v^2\)
13. \(m^2-4n^2\)
14. \(12ab-18a+6nb-9n\)
15. \(36b^2c-16ad-24b^2d+24ac\)
16. \(3m^3-6m^2n-24n^2m\)
17. \(128+54x^3\)
18. \(64m^3+27n^3\)
19. \(n^3+7n^2+10n\)
20. \(64m^3-n^3\)
21. \(27x^3-64\)
22. \(16a^2-9b^2\)
23. \(5x^2+2x\)
24. \(2x^2-10x+12\)

Answers to odd questions

1. \(6a(4c-3b)+15d(4c-3b)\)
\((4c-3b)(6a+15d)\)
\(3(4c-3b)(2a+5d)\)

3. \(-5\times -4=20\)
\(-5+-4=-9\)
\(5u^2-5uv-4uv+4v^2\)
\(5u(u-v)-4v(u-v)\)
\((u-v)(5u-4v)\)

5. \(-2(x^3-64y^3)\)
\(-2(x-4y)(x^2+4xy+16y^2)\)

7. \(2(27u^3-8)\)
\(2(3u-2)(9u^2+6u+4)\)

9. \(n(n-1)\)

11. \(x^2-3xy-xy+3y^2\)
\(x(x-3y)-y(x-3y)\)
\((x-3y)(x-y)\)

13. \((m-2n)(m+2n)\)

15. \(36b^2c-24b^2d+24ac-16ad\)
\(12b^2(3c-2d)+8a(3c-2d)\)
\((3c-2d)(12b^2+8a)\)
\(4(3c-2d)(3b^2+2a)\)

17. \(2(64+27x^3)\)
\(2(4+3x)(16-12x+9x^2)\)

19. \(5\times 2=10\)
\(5+2=7\)
\(n(n^2+7n+10)\)
\(n(n^2+5n+2n+10)\)
\(n(n(n+5)+2(n+5))\)
\(n(n+5)(n+2)\)

21. \((3x-4)(9x^2+12x+16)\)

23. \(x(5x+2)\)