Lesson 9 · from §56 of the 1910 text
Factoring: Putting It All Together
Factoring can't be boiled down to a few rules. It takes practice with the type forms until you can recognize a type just by its form. A list sorted by type feels easy; a mixed list is the real test.
I. First look for a common monomial factor, and if there is one, take out the greatest one.
II. Then decide which type the remaining polynomial is: 1. ax + ay + bx + by (grouping) 2. a^2 \pm 2ab + b^2 (perfect square) 3. x^2 + bx + c 4. ax^2 + bx + c 5. a^2 - b^2, \;a^2 + 2ab + b^2 - c^2, \;a^2 + 2ab + b^2 - c^2 - 2cd - d^2 (difference of squares) 6. a^3 \pm b^3 (sum or difference of cubes)
III. Repeat step II on each factor until every factor is prime.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Factor completely. First decide which type each one is.
x^4 + 64y^4 + 16x^2y^2
4a^2x^4 + 4a^2x^2 - 120a^2
9x^2 + 27x + 14
3x^4 + x^2 - 2
3x^3 - 75x
x^2 + 2a^2x^2 + x^2a^4
x^3 - 27
x^2 - xz + 2xy - 2yz
2ac - bc + 6ad - 3bd
25x^2y^2 + 16x^4 - 40x^3y
x^4 - 15x^3 + 56x^2
9x^6 - 6x^3 - 35
a^6 - 4096
a^{2x+2} - 1
4x^4 - 13x^2y^2 + 9y^4
216 + x^3
x^9 - 512
x^2 - 4y^2 + 9 - 6x
ah - 2ak + 2bh - 4bk
4x^4 - 25y^8
a^2 - 25c^2 + 10ab + 25b^2
y^6 - 343
9x^4 - 9x^2 - 28
9x^2y^2 + 39xy^2 - 30y^2
3x^6y^2 + 63x^3y^2 - 300y^2
12 - 15a + 16x - 20ax
2cx + 3dx - 2cy - 3dy
y^4 + z^8 - 2y^2z^4
81c^10 - 64d^10
100x^8 - 220x^4y^2 + 121y^4
27y^3 - 512
81 - 16x^8
25x^4 - 3025x^8
16c^7 - 4c^5 - 72c^3
c^3 + c^2d + 3cd^2 + 3d^3
1 + 81x^8 - 18x^4
x^4 + 12x^2 - 64
c^9 - 512c^12
h^{6a} - k^{8a}
a^2 - b^4 - 12ab + 36b^2
5h^2 + 21hk - 20k^2
x^6 - x^3 - 2
2x^2y^2 - y^4 - x^4
5h^3 - 9h^2k - 9k^3 + 5hk^2
3x^2 - 17x + 10
9x^{5n} - x^{3n}
27a - 18ab^2 - 3a^9 + 3ab^4
1 + 2x^2 - 3x^4
x^4 - x^2 - 12
16x^4 + 8x^2 - 3
1024 - 64h^3 + h^6
h^2k^2 - h^2 - k^2 + 1
12 - 2h - 4h^2
3 - y^2 + 3y^3 - y^5