### Polynomials - Multiplying and Factoring Polynomials

Multiplication of polynomials

The product of two polynomials is calculated using the following rules:

1. distributive law / rule:             a(b+c)=ab + ac

2. exponents first law:                  xaxb = xa+b

example:  Multiply: x5( 2x3 - 5x + 8)

x5( 2x3 - 5x + 8)

= x5.2x3 +x5.(-5x) +x5.8

= 2x8 -5x6 + 8x5

Multiply: (x3 + 3y2)(3x4 + 2y)

This task can be solved applying double the distributive law or FOIL rule for multiplying binomials – First Outer Inner Last

(x3 + 3y2)(3x4 + 2y) =

= x3.3x4 (First terms multiplied)

+ x3.2y (Outer terms multiplied)

+3y2.3x4 (Inner terms multiplied)

+ 3y2.2y (Last terms multiplied)

=3x7+2x3y+9x4y2+6y3

Common product forms

 Difference of two squares (x+y)(x-y) = x2-y2 Difference of two even number powers x2n-y2n=(xn-yn)(xn+yn) Square of a sum (x+ y)2= (x+ y)(x+ y) = x2+ 2xy + y2 Square of a difference (x-y)2= (x-y)(x-y)= x2-2xy + y2 Difference of two cubes, Sum of two cubes (x-y)(x2+ xy+ y2)= x3-y3 (x+y)(x2-xy+ y2)= x3+y3 Difference of two odd number powers,   Sum of two odd number powers (note alternating + and – signs) xn-yn= (x-y)(xn-1+xn-2y +xn-3y2+ …+ x2yn-3 + xyn-2+yn-1) xn+yn= (x+y)(xn-1- xn-2y +xn-3y2 - … + x2yn-3 - xyn-2+yn-1) Cube of a sum (x+y)3= (x+y)(x+y)2 = (x+ y)(x2+ 2xy + y2) = x3+ 3x2y+ 3xy2+ y3 Cube of a difference (x-y)3 = (x-y)(x-y)2 = (x-y)(x2-2xy+ y2) = x3-3x2y + 3xy2 -y3

Factoring of polynomials

Factoring is representing a polynomial as a product of two or more polynomials. Factoring reverses the distributive operation. Not all polynomials can be factored. Polynomials that can not be factored are called prime polynomials.

example:

Factor 8x4y2 + 4 x3y

8x4y2 + 4 x3y = 4x3y.2xy + 4x3y

=(2xy+1).4x3y

Note that 2xy+1 can not be factored further, therefore it is a prime polynomial

The above example is called ‘Factoring out the greatest common factor (GCF)’  GCF is the greatest factor which is common to all terms.

General steps when attempting to factor a polynomial:

1. Factor out the GCF (Greatest Common Factor) as in the above example.
2.  When there is no factor common to all terms, attempt grouping the terms which have common factor, and factor it out for each group.

example: Factor 3zx+6zy+2x+4y

Note: First two terms have common factor of 3z,

while last two terms have common factor of 2

3zx+6zy+2x+4y

= 3z(x+2y) + 2(x+2y) (Step 1)

Now we notice that (x+2y) is common factor of both groups

= (3z+2)(x+2y) (Step 2)

Note that even if Step 1 is successful, or there is a common factor for some groups of terms), it will not always yield a common factor of the groups factored in Step 1.

1. Look for difference of two squares (see common product forms above)

example: Factor 9x2 – 25x8

9x2 – 25x8 = (3x)2 – (5x4)2 = (3x – 5x4)(3x+5x4)

1. Look for sum or difference of two cubes

example: Factor x3 – 27

x3 – 27 = (x-3)(x2+3x+9)

1. Look for trinomial that can be factored, by applying a perfect square from the above common forms, or reverse FOIL afctoring.

example: Factor x2+9x+20

x2+9x+20 (make sure ordered by degree of the terms)

= x2+5x+4x+4.5 (Reverse FOIL)

= (x+4)(x+5)

Note that for reverse FOIL to be valid, the first term (9x) must be represented as sum of 2 terms (+4x +5x) with such coefficients that, as in the example, 4+5 = 9 (the original first term) and 4.5 = 20 (the original constant coefficient).