Lesson 2 · from §18 of the 1910 text
Adding Polynomials
A polynomial is an algebraic expression with two or more terms. A binomial has two terms and a trinomial has three. (We don't usually call an expression a polynomial if a letter sits under a root sign, like \sqrt{x - 3} + 4.)
Lining up columns
To add polynomials, stack them so that similar terms sit in the same column, then add each column:
\begin{array}{rrrr} 4x & -\,6b & -\,a^2c & \\ & 3b & & \\ -3x & & -\,7a^2c & +\,10 \\ 5x & +\,3b & & -\,6 \\ \hline 6x & & -\,8a^2c & +\,4 \end{array}
Rule. Write similar terms in the same column. Find the algebraic sum of each column, and write the results one after another with their proper signs.
Quick check
Add 3x - 2y and -x + 5y - 4.
Checking your work
A check is a second operation that tests the first. In arithmetic you check 132 \div 6 = 22 by multiplying 22 \cdot 6 = 132.
Since the letters stand for any numbers, you can check a sum by giving them simple values, say a = b = c = 1:
\begin{array}{rcl} 5a - 3b - 3c & \to & 5 - 3 - 3 = -1 \\ 2a - 5b + 6c & \to & 2 - 5 + 6 = 3 \\ -3a + 2b - 4c & \to & -3 + 2 - 4 = -5 \\ \hline 4a - 6b - c & \to & 4 - 6 - 1 = -3 \end{array}
The polynomials' values add to -1 + 3 - 5 = -3, and the sum is also -3, so the addition checks. A check like this catches most mistakes, though not all: two mistakes can cancel. (The wrong sum 4a - 5b - 2c also gives -3 when every letter is 1.) Giving each letter a different number makes the check more reliable.
Quick check
Check: what is the value of 4a - 6b - c when a = 2, b = 1, c = 3?
Polynomial coefficients
The sum of 5x and 2x could be written (5+2)x, but you'd simply write 7x. The sum of 5x and ax can't be squeezed into one coefficient, so it's written (5 + a)x. Likewise ax - 3x = (a - 3)x and ax - bx + x = (a - b + 1)x.
Quick check
Write mx + 2x with a polynomial coefficient.
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.
Add the polynomials, then check your sum by letting every letter equal 1.
Add x + y + z, x - 2y + 3z, and 3x + 4y - 7z.
Add 2x + 5y - z, 3x - 8y + 6z, and x - y - z.
Add 3x + 5y, 4x - 7y + 6z, and 3x - 3y - 3z.
Add 7x - y + 3z, 5x - 4z, and 2x + 6y - 5z.
Add 4x - 3y - 5z, 6y - 2z, and 7x - 6y - 4z.
Add x + 2z + 3y, y - 3z + x, and z - 2x - 4y.
Add 5x - y + 3z, 2y - 11z + x, and 9z - 7y.
Add 8a - 7b - 6c, 5c - 4a - 3b, and 3b + 7c.
Add 9ac - bc, 8ab - 4ac, and -12ab - ac.
Add a^2 - 4a + 10, 5a - 6a^2 + 4, and 3a - 16 + 2a^2.
Add 9 - a^2 + a, -7 + 6a^2 - 4a, and 5a^2 - a.
Add \tfrac12 x - \tfrac23 y + \tfrac34 z, x - y + 2z, and \tfrac13 x - \tfrac12 y + \tfrac1{10} z.
Add \tfrac14 x - \tfrac3{10} z, \tfrac18 x + \tfrac49 y, and \tfrac32 z - \tfrac53 y.
Add x - 3(x-y) + z, 5 - 10x + 4(x-y), and -2(x-y) + 6.
Add a + d + 2(b-c), 7(b-c) + 6d - 12a, and 11a - 5(b-c).
Combine similar terms.
Combine similar terms: 7a^2 - 13b^2 + 12c^2 + 15b^2 - a^2 - 7c^2 + 3b^2 + 5c^2
Combine similar terms: x^2 - 2xy + y^2 - 4xy - 4x^2 - y^2 - 16x^2 - 8xy + y^2 + y^3
Combine similar terms: 5ab - a^2 + b^2 - 4ab - 9b^2 + 5a^2 - 2ab + 2b^2
Combine similar terms: 3x^2 - 6x + 11 - 4x - 8 - 3x^2 + 5x - 16 + 13x
Combine similar terms: 12c^2 - 10bc + 8b^2 - bc - 6b^2 - c^2 + c^3 - 11c^2
Combine similar terms: 4x^2y - xy + y^2 - 3xy + 4xy^2 - 2x^2y + 4y^2 - 3xy + x^2y
Combine similar terms: \tfrac12 a + \tfrac13 b - \tfrac14 c - c + \tfrac12 b - \tfrac23 a - b + \tfrac12 c - \tfrac34 a + 7
Write each with a polynomial coefficient.
Write with a polynomial coefficient: ax + bx + cx
Write with a polynomial coefficient: 2ax - 3x + bx
Write with a polynomial coefficient: 3ax - 4cx + x
Write with a polynomial coefficient: 3ax^2 - bx^2 - x^2 + a^2x^2
Write with a polynomial coefficient: by - 4cy - y - 4by
Write with a polynomial coefficient: a(b+c) + 3(b+c)
Write with a polynomial coefficient: 4(a-x) - 3b(a-x)
Write with a polynomial coefficient: 8a(a+3b) - 1(a+3b)
Write with a polynomial coefficient: 7b(x^2+y^2) - a(x^2+y^2) + (x^2+y^2)