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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