Multiplying Polynomials

Objectives Multiply a polynomial by a monomial. Multiply a polynomial by a polynomial. Objectives

The Distributive Property Review The Distributive Property Look at the following expression: 3(x + 7) This expression is the sum of x and 7 multiplied by 3. (3 • x) + (3 • 7) 3x + 21 To simplify this expression we can distribute the multiplication by 3 to each number in the sum.

Review Whenever we multiply two numbers, we are putting the distributive property to work. We can rewrite 23 as (20 + 3) then the problem would look like 7(20 + 3). 7(23) Using the distributive property: (7 • 20) + (7 • 3) = 140 + 21 = 161 When we learn to multiply multi-digit numbers, we do the same thing in a vertical format.

Review 7 • 3 = 21. Keep the 1 in the ones position then carry the 2 into the tens position. 2 23 x____ 7 7 • 2 = 14. Add the 2 from before and we get 16. 16 1 What we’ve really done in the second step, is multiply 7 by 20, then add the 20 left over from the first step to get 160. We add this to the 1 to get 161.

Multiplying a Polynomial by a Monomial Multiply: 3xy(2x + y) This problem is just like the review problems except for a few more variables. To multiply we need to distribute the 3xy over the addition. 3xy(2x + y) = (3xy • 2x) + (3xy • y) = 6x2y + 3xy2 Then use the order of operations and the properties of exponents to simplify.

Multiplying a Polynomial by a Monomial We can also multiply a polynomial and a monomial using a vertical format in the same way we would multiply two numbers. Multiply: 7x2(2xy – 3x2) 2xy – 3x2 Align the terms vertically with the monomial under the polynomial. x________ 7x2 14x3y – 21x4 Now multiply each term in the polynomial by the monomial. Keep track of negative signs.

Multiplying a Polynomial by a Polynomial To multiply a polynomial by another polynomial we use the distributive property as we did before. Multiply: (x + 3)(x – 2) (x + 3) Line up the terms by degree. x________ (x – 2) Multiply in the same way you would multiply two 2-digit numbers. 2x – 6 _________ x2 + 3x + 0 x2 + 5x – 6 Remember that we could use a vertical format when multiplying a polynomial by monomial. We can do the same here.

Multiplying a Polynomial by a Polynomial To multiply the problem below, we have distributed each term in one of the polynomials to each term in the other polynomial. Here is another example. Multiply: (x + 3)(x – 2) (x2 – 3x + 2)(x2 – 3) (x + 3) (x2 – 3x + 2) x________ (x – 2) Line up like terms. 2x – 6 x____________ (x2 – 3) x2 _________ + 3x + 0 – 3x2 + 9x – 6 x2 + 5x – 6 x4 __________________ – 3x3 + 2x2 + 0x + 0 x4 – 3x3 – 1x2 + 9x – 6

Multiplying a Polynomial by a Polynomial It is also advantageous to multiply polynomials without rewriting them in a vertical format. Though the format does not change, we must still distribute each term of one polynomial to each term of the other polynomial. Each term in (x+2) is distributed to each term in (x – 5). Multiply: (x + 2)(x – 5)

Multiplying a Polynomial by a Polynomial Multiply the First terms. O Multiply the Outside terms. F (x + 2)(x – 5) Multiply the Inside terms. Multiply the Last terms. I After you multiply, collect like terms. L This pattern for multiplying polynomials is called FOIL.

Multiplying a Polynomial by a Polynomial Example: (x – 6)(2x + 1) x(2x) + x(1) – (6)2x – 6(1) 2x2 + x – 12x – 6 2x2 – 11x – 6

You Try It!! 1. 2x2(3xy + 7x – 2y) 2. (x + 4)(x – 3)

Problem One 2x2(3xy + 7x – 2y) 2x2(3xy + 7x – 2y) 2x2(3xy) + 2x2(7x) + 2x2(–2y) Problem One 6x3y + 14x2 – 4x2y

Problem Two (x + 4)(x – 3) (x + 4)(x – 3) x(x) + x(–3) + 4(x) + 4(–3)

Problem Three (2y – 3x)(y – 2) (2y – 3x)(y – 2) 2y(y) + 2y(–2) + (–3x)(y) + (–3x)(–2) Problem Three 2y2 – 4y – 3xy + 6x