Lesson 1 · from §58 of the 1910 text
Highest Common Factor
Degree
The degree of a monomial is the sum of the exponents of its letters: ax^3 is of the fourth degree, and 4a^3x^3y^2 is of the eighth. The degree of a polynomial is the degree of its highest term, so 6axy^2 + 4a^3x^4 - 3ax^2yz is of the seventh degree.
The H.C.F.
The highest common factor (H.C.F.) of two or more expressions is the expression of highest degree, with the greatest numerical coefficient, that divides each of them exactly. The H.C.F. of 36a^2b^2 and 48a^3b is 12a^2b.
Example. Find the H.C.F. of 24x^3y^2z, 48x^4y^6z^3, and 72x^5y^5z^2. Factor: 2^3\cdot3\,x^3y^2z, \;2^4\cdot3\,x^4y^6z^3, \;2^3\cdot3^2x^5y^5z^2. Take the lowest power of each factor that appears in all of them: 2^3\cdot3\,x^3y^2z = 24x^3y^2z.
Example. 2x^4 - 12x^3 + 18x^2 = 2x^2(x - 3)^2 and 4x^5 - 36x^3 = 2^2x^3(x - 3)(x + 3). Their H.C.F. is 2x^2(x - 3) = 2x^3 - 6x^2.
Rule. Separate each expression into its prime factors. Then find the product of such factors as occur in every expression, using each the least number of times it occurs in any one expression.
Quick check
36a^2b^2 and 48a^3b
From the history of algebra
Euclid's Elements (about 300 BC) gives a method for the greatest common divisor of two numbers that never needs factoring: keep dividing and taking remainders. It's among the oldest algorithms still in daily use, and it works for polynomials too, which is how computers find an H.C.F. today.
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.
Numbers first.
12 and 18
24 and 56
96 and 144
84 and 196
125 and 225
64, 96 and 256
90, 108 and 324
Monomials. Leave your answer as a single term like 6x^3.
12x^4 and 18x^3
16x^2y^4 and 24x^3y^5
27c^2d and 21cd^4
32a^3bc^4 and 48a^2b^2d
125m^7n^5p and 100ng^4
18h^2k^2, 36h^4k and 24h^3k^3
9xy^4, 54x^5y and 15x^4y^5
27a^4b^5c^2, 54a^3b^2d and 81a^2b^2e^3
Polynomials. Factor each one first. You can type the H.C.F. factored or multiplied out.
x^2-9 and x^2-5x+6
x^2+3x-10 and x^2+6x+5
x^3-4x and x^3-8x^2+12x
2c^3+12c^2+18c and c^3-2c^2-15c
8+y^3 and y^2+4y+4
x^4-2x^2+1 and x^2-2x+1
ab+3b+ac+3c and 2ab+6b-2ac-6c
c^2+3cd+2d^2, c^2+5cd+6d^2 and c^2+cd-2d^2