Lesson 2 · from §43 of the 1910 text
Sum Times Difference
Multiplying a + b by a - b makes the middle terms cancel: (a + b)(a - b) = a^2 - b^2.
III. The product of the sum and the difference of two terms equals the difference of their squares, taken in the same order as the difference of the terms.
That's a great mental-math trick: 22 \cdot 18 = (20 + 2)(20 - 2) = 400 - 4 = 396.
Quick check
31 \cdot 29
Read backward, any difference of two squares splits into a sum times a difference: c^2 - d^2 = (c + d)(c - d). Again, a "term" can be a whole group: [a + (x + y)][a - (x + y)] = a^2 - (x + y)^2 = a^2 - x^2 - 2xy - y^2.
Quick check
Write x^2 - 49 as the product of two binomials.
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.
Expand by the formula.
(6 + 3)(6 - 3)
(8 + 4)(8 - 4)
(9 - 2)(9 + 2)
(10 + 4)(10 - 4)
(7 + 2)(7 - 2)
(12 + 3)(12 - 3)
22 \cdot 18
35 \cdot 25
33 \cdot 27
36 \cdot 44
35 \cdot 45
52 \cdot 48
65 \cdot 75
72 \cdot 68
75 \cdot 85
97 \cdot 103
(x + 3)(x - 3)
(a + 5)(a - 5)
(2x + 4)(2x - 4)
(3n + 5)(3n - 5)
(x + y)(x - y)
(x - a)(x + a)
(x - 1)(x + 1)
(a + 2)(a - 2)
(a^2 + 3)(a^2 - 3)
(4 + y^2)(4 - y^2)
(4 - x)(x + 4)
(2c + a)(2c - a)
(3a + b)(b - 3a)
(4b + 2c)(4b - 2c)
(3xy - 2)(3xy + 2)
(4ab - 3)(4ab + 3)
(a^4 - b^3)(a^4 + b^3)
(a^6 + a^3)(-a^3 + a^6)
(x^3 - 2y)(x^3 + 2y)
(4ab - a^2)(4ab + a^2)
(3cd^2 + 2d)(-2d + 3cd^2)
(6cd + 3)(-3 + 6cd)
(4xy + 2y)(2y - 4xy)
(3abc - 2bc)(2bc + 3abc)
Divide.
\dfrac{a^2 - b^2}{a + b}
\dfrac{c^2 - d^2}{c - d}
\dfrac{9 - b^2}{3 + b}
\dfrac{36 - a^2}{6 - a}
\dfrac{9x^2 - 16}{4 + 3x}
\dfrac{x^4 - 1}{x^2 - 1}
Find a binomial divisor of each.
x^2 - y^2
x^2 - 1
4x^2 - 9
25 - 16x^2
16 - x^8
y^6 - 4
Write each as the product of two binomials.
c^2 - d^2
n^2 - 4
n^2 - 16
9 - 4a^2
36b^2 - 1
a^2 - 9
25 - 4n^2
100 - 9x^2
Multiply, treating a group as one term.
[(x + y) + 1][(x + y) - 1]
(x + a + 3)(x + a - 3)
[(x - a) + 3][(x - a) - 3]
(x + 4 + c)(x + 4 - c)
(2a - b + c)(2a - b - c)
[x + (b + c)][x - (b + c)]
[x + (b - c)][x - (b - c)]
[3 + (x - y)][3 - (x - y)]
[4x + (2y - x)][4x - (2y - x)]
[10 - (a - 5)][10 + (a - 5)]
Write each as the product of two binomials.
49x^2 - 1
64x^2 - 25
(a + b)^2 - 1
(x - y)^2 - 4
(2x - 1)^2 - a^2
(3y - z)^2 - b^2
b^2 - (x + y)^2
b^2 - (x - y)^2
c^2 - (x + c)^2
4 - (x - 2)^2