Factoring Polynomials

Degree of a product is the sum of degrees of the factors Let’s take a look at some products of polynomials that we saw before in the chapter on “Basics of Polynomials”: The leading term of (2x2 − 5x)(−7x + 4) is −14x3. This is an example of a degree 2 and a degree 1 polynomial whose product equals 3. Notice that 2 + 1 = 3 The product 5(x− 2)(x+3)(x2 +3x− 7) is a degree 4 polynomial because its leading term is 5x4. The degrees of 5, (x− 2), (x+ 3), and (x2 + 3x− 7) are 0, 1, 1, and 2, respectively. Notice that 0 + 1 + 1 + 2 = 4. The degrees of (2x3 − 7), (x5 − 3x+ 5), (x− 1), and (5x7 + 6x− 9) are 3, 5, 1, and 7, respectively. The degree of their product, (2x − 7)(x − 3x+ 5)(x− 1)(5x + 6x− 9), equals 16 since its leading term is 10x16. Once again, we have that the sum of the degrees of the factors equals the degree of the product: 3+5+1+7 = 16. These three examples suggest a general pattern that always holds for factored polynomials (as long as the factored polynomial does not equal 0):