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Multiplying Polynomials

What to Do How to Do It
1. The most basic property of algebra necessary to multiplying polynomials is the distributive property.   A(B + C) = AB + AC
2. This can be extended to several terms inside the parenthesis: (The three dots ... mean and so on in the pattern.)   → A(B + C + D + ... )

   = AB + AC + AD + ...

3. Both multiplicands may be binomials, in which case the property is usually referred to as the “double distributive property”.   (A + B)(C + D)

  = A(C + D) + B(C + D)

  = AC + AD + BC + BD

4. Consider the following examples:  
a) 2(x + 3) =   2(x + 3)

  = 2x + 2·3

  = 2x + 6

b) 3x(x-1) =   3x(x - 1)

  = 3x·x + 3x·(-1)

  = 3x2 - 3x

c) - 5(x2 + 2x - 3) =   - 5(x2 + 2x - 3)

  = (- 5)(x2) + (- 5)(2x) + (- 5)(- 3)

  = - 5x2 - 10x + 15

d) (x + a)(y + b) =   → (x + a)(y + b)

  = x(y + b) + a(y + b)

  = xy + xb + ay + ab

e) (x + 2)(2x + 3) =

With practice we can do the second line in our head and go directly to the third line.

  (x + 2) (2x + 3)

  = x (2x + 3) + 2(2x + 3)

  = x·2x + x·3 + 2·2x + 2·3

  = 2x2 + 3x + 4x + 6

  = 2x2 + 7x + 6

 

f) (2x + 3)(2x + 3) =

Later, skip this second line and →

go directly to this third line. →

  (2x + 3) (2x + 3)

  = 2x·2x + 2x·3 + 3·2x + 3·3

  = 4x2 + 6x + 6x + 9    

  = 4x2 + 12x + 9

g) (2x + 3)(2x - 3) =

Note the “equal and opposites” →

cancel out here.

  (2x + 3) (2x - 3)

  = 2x·2x + 2x·(- 3) + 3·2x + 3·(- 3)

  = 4x2 - 6x + 6x - 9

  = 4x2 - 9

     

 
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