9.1 Operations with Polynomials

1. 9.1 Operations with Polynomials ©2001 by R. Villar All Rights Reserved

2. Operations with Polynomials Term:each monomial in a polynomial. Degree: degree of the monomial with the greatest degree (exponents). The degree of the polynomial above is 3 Polynomial: a monomial or the sum or difference of monomials.

3. 1 1 Example: Find the degree of Degree: 3 4 3 2 The degree of the polynomial is 4 Like Terms:terms with the same variables(s) to the same power(s). Example: Simplify by combining like terms3mn – 12m2 + 6mn + 3m2 – 7mn – 9m2 + 2mn

4. Multiply: –2a2b(3a3 + 5b2 – 2) –6a5b – 10a2b3 + 4a2b Multiply: (x + 5)(x – 3) F O I L first outer inner last x2 –3x + 5x – 15 x2 +2x – 15

5. Example Simplify (a + 2)(a2 – 3a + 5) FOIL will not work here because this is abinomialmultiplied by atrinomial.(FOIL is only for binomials by binomials) Instead, you must distribute twice. a(a2 – 3a + 5)+ 2(a2 – 3a + 5) a3 – 3a2 + 5a + 2a2 – 6a + 10 a3 – a2 – a + 10

6. Recall, how do you square a binomial? Example: Simplify (x + 4)2 x2 +4x + 4x + 16 x2 + 8x + 16 Here’s the shortcut... Square of a binomial: (a + b)2 a2 + 2ab + b2

7. How do you cube a binomial? Example: Simplify (x + 4)3 This is the same as (x + 4)(x + 4)(x + 4) = (x + 4)(x2 + 8x + 16) = x(x2 + 8x + 16) + 4(x2 + 8x + 16) = x3 + 8x2 + 16x + 4x2 + 32x + 64 = x3 + 12x2 + 48x + 64 Here’s the shortcut... Cube of a binomial: (a + b)3 a3 + 3a2b + 3ab2 + b3 Example: Simplify (x – 3)3 = x3 – 9x2 + 27x – 27