Exponents and Polynomials - XYZ Custom Plus

9.3 Multiplying Polynomials: An Introduction583Example 9Expand and multiply: (3x − 7) 2Solution We know that (3x − 7) 2 = (3x − 7)(3x − 7). It will be easier to applythe distributive property to this last expression if we think of the second 3x − 7 as3x + (−7). In doing so we will also be less likely to make a mistake in our signs.(Try the problem without changing subtraction to addition of the opposite, andsee how your answer compares to the answer in this example.)9. Expand and multiply: (3x − 5) 2( 3 x − 7)(3x − 7) = (3x − 7)[3x + (−7)]= (3x − 7) ⋅ 3x + (3x − 7)(−7)= 3x ⋅ 3x − 7 ⋅ 3x + 3x(−7) − 7(−7)= 9x 2 − 21x − 21x + 49= 9x 2 − 42x + 49CMultiplying Polynomials GeometricallySuppose we have a rectangle with length x + 3 and width x + 2. Remember, theletter x is used to represent a number, so x + 3 and x + 2 are just numbers. Here isa diagram:x3x2The area of the whole rectangle is the length times the width, orTotal area = (x + 3)(x + 2)But we can also find the total area by first finding the area of each smaller rectangleand then adding these smaller areas together. The area of each rectangle isits length times its width, as shown in the following diagram:xxx 233x2 2x6Because the total area (x + 3)(x + 2) must be the same as the sum of the smallerareas, we have:( x + 3)(x + 2) = x 2 + 2x + 3x + 6= x 2 + 5x + 6 Add 2x and 3x to get 5xThe polynomial x 2 + 5x + 6 is the product of the two polynomials x + 3 and x + 2.Here are some more examples.Answers9. 9x 2 − 30x + 25