Lesson 3 · from §50 of the 1910 text
Factoring by Grouping
The type form is ax + ay + bx + by. Group it: ax + ay + bx + by = a(x + y) + b(x + y) = (x + y)(a + b).
Rule. Arrange the terms in groups so that each group has a monomial factor that leaves the same expression in parentheses. Then that expression is one factor, and what multiplies it is the other.
These usually have four, six, or eight terms. Watch signs: (a - 3b)(c - d) and (3b - a)(d - c) are the same product, just with both signs flipped, so either is correct.
Example. 4x^3 - 3xy - 8x^2y + 6y^2 = x(4x^2 - 3y) - 2y(4x^2 - 3y) = (4x^2 - 3y)(x - 2y).
Factoring is harder than multiplying, since you're only given the product. There's no single operation that guarantees you've found the prime factors; insight and practice do that. But you can always check by multiplying your factors back together.
Quick check
xy + 3x + 2y + 6
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.
2(x + y) + a(x + y)
b(x + 2y) + (x + 2y)
c(c - a) - a(c - a)
2c(3b - d) + 3a(3b - d)
6d(c + x) - (c + x) + 4k(c + x)
-5(2a - 3b) + 7a(2a - 3b) - 3b(2a - 3b)
k(a - b) + 3(b - a)
h(x - y) + k(y - x)
5x(c - 2d) - 6(2d - c)
3y(k - 4h) + (4h - k)
2c(4k - 3c) - 5h(3c - 4k)
3y(5x - b) - 7(b - 5x) + 6z(5x - b)
ab + bx + ac + cx
ax - bx + a - b
4x^3 - 3xy - 8x^2y + 6y^2
ax + bx + ay + by
ch + h^2 + cx + hx
bh - ch + bk - ck
bx + xz - ab - az
ax + 3a - bx - 3b
2x^2 + 10xy - 4x - 20y
6ab - 2ac + 3b - c
36ax + 45ac - 4x - 5c
6a^3 + 14ab - 15a^2y - 35by
3ax - ay - 6bx + 2by
12ax - 6ay - 50cx + 25cy
2h^3 - 3h^2k - 10h + 15k
10dh - 45hk - 22cd + 99ck
28hx + 9ky - 21hy - 12kx
-2ax + 7a^2 + 16bx - 56ab
6hk + 15xy - 10ky - 9hx
-21ax + 12cx - 4cd + 7ad
7gy - 77hy - cg + 11ch
5a^3 + 10a - 5a^2 - 10
3a - 5ax^3 - 6ax + 10ax^4
4abxy - 24dxy - 3abgh + 18dgh
8acxy - 20bx^2y - 6abc^2 + 15b^2cx
ax + bx + cx + ay + by + cy
cg + 2ch + ck - gx - 2hx - kx
3ab - bx + 2by - 3ac + cx - 2cy
4ax - ac + 2ay + 12bx - 3bc + 6by
ax - 6cy - 2bx + 4by + 3cx - 2ay
-2xz + 6ay + cz + 4xy - 3az - 2cy