Multiplying Polynomials Using the Distributive Property Monomial times

Multiplying Polynomials

Using the Distributive Property Monomial times Polynomial: distribute the monomial (2 x) 1. 2 x(4 x 3 -6) to each term 2 x(4 x 3)+2 x(-6) 8 x 4 -12 x 2. 5 x(3 x 2 -2 x+1) (distribute the 5 x to each term) 5 x(3 x 2)+5 x(-2 x)+5 x(1) 15 x 3 -10 x 2+5 x 3. 6 x 2(5 x 2+3 x-9) 30 x 4+18 x 3 -54 x 2

Use the Distributive Property Binomial times Polynomial: (x-2)(5+3 x-x 2) Split the binomial up into its two terms and distribute them each as a monomial to each term in the polynomial =x(5+3 x-x 2)+(-2)(5+3 x-x 2) =5 x + 3 x 2 - x 3 – 10 - 6 x + 2 x 2 =-x + 5 x 2 - x 3 – 10 =-x 3+5 x 2 -x-10 Combine like terms Write in standard form

Three Ways to Multiply a Binomial times Binomial: 1. Use the Distributive Property (x + 1)(3 x – 2) Just like the last example, split the binomial up into its two terms and distribute them both =x(3 x-2)+(1)(3 x-2) =3 x 2 - 2 x + 3 x -2 =3 x 2+x-2 2. Area Model 3. FOIL Method

Multiply Using Area Model: (x + 1)(3 x – 2) w Assign each binomial to a side of the rectangle. w Our goal is to calculate the area of the full rectangle by splitting the side lengths into more manageable parts. w Use “x + 1” and subdivide the side of the rectangle. – (split it up into its two terms) w Use “ 3 x – 2” and subdivide the top of the rectangle. – (split it up into its two terms) w Multiply the area of each smaller section 3 x - 2 x 3 x 2 + 1 3 x -2 Now add up all of the areas: 3 x 2+3 x– 2 x-2 =3 x 2+x-2

Multiply Using Area Model: w w w -9 2 x Multiply (3 x + 1)(2 x – 9) 3 x(2 x) = 6 x 2 3 x 6 x 2 -27 x 3 x(-9) = -27 x 1(2 x) = 2 x 1 2 x -9 1(-9) = -9 Next add up all of the areas: 6 x 2 – 27 x + 2 x - 9 = 6 x 2 - 25 x - 9

Multiply Using Area Model: w You will get the same answer using the distributive property or the FOIL Method. The area model is just another way to perform the same operation w Multiply (5 x - 4)(-3 x + 7) -3 x 7 5 x -15 x 2 35 x -4 12 x -28 Now add up all of the areas: -15 x 2 + 47 x - 28

FOIL Method F first terms O outer terms I inner terms L last terms A short-cut method to multiply two binomials together

Multiply Using FOIL: -Multiply the First terms together -Multiply the Outer terms -Multiply the Inner terms -Multiply the Last terms -Combine like terms at the end (2 x-3) (3 x+4) = 2 x 3 x + 2 x 4 +(-3) 3 x +(-3) 4 = 6 x 2 + =6 x 2 -x-12 8 x - 9 x - 12

Practice FOIL: (5 x+3) (4 x-6) 20 x 2 – 30 x + 12 x – 18 20 x 2 – 18 x - 18