The prime factorization of an integer is a term with prime numbers multiplied together giving the same value as the integer. A prime number is an irreducible factor. Similarly, a polynomial expression can be represented as a product of irreducible polynomials. These are polynomials with coefficients in the set of real numbers that cannot be further reduced into other factors aside from the number 1 and itself. The need to factor a polynomial into its prime factors occurs frequently in mathematics. A general factoring strategy to do this follows.
Remove factors common to all terms.
Grouping common factors:2x4 + 16x2 + 8x = 2x(x3 + 8x + 4) 6(x2 + 1)3(3x - 1)2 + 8x(x2 + 1)2(3x - 1)3 = 2(x2 + 1)2(3x - 1)2[3(x2 + 1) + 4x(3x - 1)] = 6(x2 + 1)3(3x - 1)2(3x2 + 3 + 12x2 - 4x) = 6(x2 + 1)3(3x - 1)2(15x2 - 4x + 3)
Reverse FOIL is derived from the following binomial expansions:6x2 + 8xy - 3x - 4y = 2x(3x + 4y) - (3x + 4y) = (2x - 1)(3x + 4y)
To factor the expression x2 + 9x + 20, find two factors of 20 whose sum is 9; 4×5 = 20. Then,x2 + (a + b)x + ab = (x + a)(x + b) acx2 + (bc + ad)xy + bdy2 = (ax + by)(cx + dy)
Factor the expression 8x2 + 22xy + 15y2. Find two factors of 8x15 = 120 whose sum is 22. 12x10 = 120, 12 + 10 = 22. Then, chose values for a, b, c, d so that: ac = 8, bd = 15, bc = 12, ad = 10, (some trial and error may be necessary). Let, a = 2, c = 4, b = 3, d = 5; then,x2 + 9x + 20 = (x + 4)(x + 5)
Look for similarity to a special product form. Some special product forms are given below:8x2 + 22xy + 15y2 = (2x + 3y)(4x + 5y)
For example, factor the polynomial 4x2 - 9. This is a difference of two squares with a = 2x, b = 3. It can be factored into (2x + 3)(2x - 3).
a2 - b2 = (a + b)(a - b) difference of two squares a2 + 2ab + b2 = (a + b)2 square of a sum a2 - 2ab + b2 = (a - b)2 square of a difference a3 + b3 = (a + b)(a2 - ab + b2) sum of two cubes a3 - b3 = (a - b)(a2 + ab + b2) difference of two cubes a3 + 3a2b + 3ab2 + b3 = (a + b)3 cube of a sum a3 - 3a2b + 3ab2 - b3 = (a - b)3 cube of a difference
Another example, factor the polynomial 8x3 + 27y3. This is a sum of two cubes with a = 2x, b = 3y. It can be factored into (2x + 3)(4x2 - 6xy + 9y2).