Lesson 4 · from §45, 46 of the 1910 text
Squares of Polynomials, Cubes of Binomials
Multiplying a + b - c by itself gives (a + b - c)^2 = a^2 + b^2 + c^2 + 2ab - 2ac - 2bc.
V. The square of any polynomial equals the sum of the squares of its terms, plus twice the product of each term with each term that follows it.
Quick check
(x + y + 2)^2
The cube of a binomial
Multiply (a + b)^2 = a^2 + 2ab + b^2 by a + b once more: (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3, \qquad (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3. In words: the cube of a binomial is the cube of the first term, plus three times the square of the first times the second, plus three times the first times the square of the second, plus the cube of the second. With a minus sign, the signs alternate.
Quick check
(a + 2)^3
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.
Square by rule V.
(a + b + c)^2
(a + b + 1)^2
(a + b + 2)^2
(2a + b - c)^2
(a - b - c)^2
(2x + y - 1)^2
(4x + y - 2)^2
(a - bc + d)^2
(3a - 3b + 1)^2
(a - b + c - d)^2
(x + y - a + 1)^2
(a - y + b - 3)^2
Cube each binomial by the formula.
(x + y)^3
(x - y)^3
(x + 1)^3
(x - 1)^3
(x + 2)^3
(x - 2)^3
(x - 3)^3
(x + 3)^3
(2x + y)^3
(x - 2y)^3
(2x - 1)^3
(3x + 2)^3
(3x - 2y)^3
(4x + 3y)^3
(5x - 2y)^3