# Distributive Property (Multiplying a monomial by a polynomial)

 The distributive property is written as follows: a(b+c)=ab+ac This property has many applications, but it is particularly valuable to help us multiply a monomial by a polynomial. For example, x(3x+5). Since there are variables involved, we cannot add what is in parenthesis first (remember, 3x and 5 are not like terms). Instead, we will use the distributive property to multiply. The best way to use the distributive property is to remember these three steps: 1) Multiply the outside term by the first term in parenthesis 2) Put a plus sign 3) Multiply the outside term by the second term in parenthesis Let's look at few examples 1) x(3x+5)=3x2+5x Step 1: Multiply the outside term by the first term in parenthesis x.3x=3x2 Step 2: Put a plus sign Step 3: Multiply the outside term by the second term in parenthesis: x.5=5x The answer cannot be simplified because there are no like terms, and it is in standard form, so we are finished. Final answer: 3x2+5x 2) 2y(y-8)=2y2+(-16y)=2y2-16y Step 1: Multiply the outside term by the first term in parenthesis 2y.y=2y2 Step 2: Put a plus sign Step 3: Multiply the outside term by the second term in parenthesis: 2y(-8)=-16y This could be our final answer, but the plus sign isn't needed in this problem, so we could rewrite it as 2y2-16y. 3) 3x2 (5x2-4x+2)=15x4+(-12x3 )+6x2=15x4-12x3+6x2 Step 1: Multiply the outside term by the first term in parenthesis 3x2.5x2=15x4 Step 2: Put a plus sign Step 3: Multiply the outside term by the second term in parenthesis: 3x2 (-4x)=-12x3 This problem has a third term inside parenthesis, so we will just continue the pattern: Step 4: Put a plus sign Step 5: Multiply the outside term by the third term in parenthesis: 3x2 (2)=6x2 This could be our final answer, but the first plus sign isn't needed in this problem, so we could rewrite it as 15x4-12x3+6x2. Practice: Multiply (distribute) the following: 1) 3(y+5) 2) 4x(x-2) 3) -4(2y-6) 4) 3a(a2-4) 5) 7x(x2+5x-8) Answers: 1) 3y+15 2) 4x2-8x 3) -8y+24 4) 3a3-12a 5) 7x3+35x2-56x