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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 â€œdouble distributive propertyâ€.	→ (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 + 2Â·3 = 2x + 6
b) 3x(x-1) =	→ 3x(x - 1) = 3xÂ·x + 3xÂ·(-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) = xÂ·2x + xÂ·3 + 2Â·2x + 2Â·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) = 2xÂ·2x + 2xÂ·3 + 3Â·2x + 3Â·3 = 4x² + 6x + 6x + 9 = 4x² + 12x + 9
g) (2x + 3)(2x - 3) = Note the â€œequal and oppositesâ€ → cancel out here.	→ (2x + 3) (2x - 3) = 2xÂ·2x + 2xÂ·(- 3) + 3Â·2x + 3Â·(- 3) = 4x² - 6x + 6x - 9 = 4x² - 9