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)=3x^{2}+5x Step 1: Multiply the outside term by the first term in parenthesis x.3x=3x^{2} 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: 3x^{2}+5x 2) 2y(y-8)=2y^{2}+(-16y)=2y^{2}-16y Step 1: Multiply the outside term by the first term in parenthesis 2y.y=2y^{2} 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 2y^{2}-16y. 3) 3x^{2} (5x^{2}-4x+2)=15x^{4}+(-12x^{3} )+6x^{2}=15x^{4}-12x^{3}+6x^{2} Step 1: Multiply the outside term by the first term in parenthesis 3x^{2}.5x^{2}=15x^{4} Step 2: Put a plus sign Step 3: Multiply the outside term by the second term in parenthesis: 3x^{2} (-4x)=-12x^{3}
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: 3x^{2} (2)=6x^{2} This could be our final answer, but the first plus sign isn't needed in this problem, so we could rewrite it as 15x^{4}-12x^{3}+6x^{2}. Practice: Multiply (distribute) the following: 1) 3(y+5) 2) 4x(x-2) 3) -4(2y-6) 4) 3a(a^{2}-4) 5) 7x(x^{2}+5x-8) Answers: 1) 3y+15 2) 4x^{2}-8x 3) -8y+24 4) 3a^{3}-12a 5) 7x^{3}+35x^{2}-56x |
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