DAY 132 POLYNOMIAL OPERATIONS VOCABULARY Multiply Polynomials Check

DAY 132 –POLYNOMIAL OPERATIONS

VOCABULARY Multiply Polynomials – Check out the following products of polynomials: Multiplying a Polynomial by a Monomial – Use the Distributive Property and the Rules for Multiplying Exponential Expressions. Example:

VOCABULARY Multiplying 2 Polynomials – Multiply each term of one polynomial by each term of the other polynomial and then Combine like terms. Example:

OBJECTIVES 1. Multiply a polynomial by a monomial. 2. Multiply a polynomial by a polynomial.

REVIEW: DISTRIBUTIVE PROPERTY Look at the following expression: 3(x + 7) (3 • x) This expression is the sum of x and 7 multiplied by 3. + (3 • 7) 3 x + 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. 7(23) We can rewrite 23 as (20 + 3) then the problem would look like 7(20 + 3). 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.

2 23 x____7 16 1 7 • 3 = 21. Keep the 1 in the ones position then carry the 2 into the tens position. 7 • 2 = 14. Add the 2 from before and we get 16. 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: 3 xy(2 x + y) This problem is just like the review problems except for a few more variables. To multiply we need to distribute the 3 xy over the addition. 3 xy(2 x + y) = (3 xy • 2 x) + (3 xy • y) = 6 x 2 y + 3 xy 2 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: 7 x 2(2 xy – 3 x 2) 2 xy – 3 x 2 x____ 7 x 2 14 x 3 y – 21 x 2 Keep track of negative signs. Align the terms vertically with the monomial under the polynomial. Now multiply each term in the polynomial by the monomial.

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) x____ (x – 2) 2 x – 6 x 2 + 3 x + 0 _____ Line up the terms by degree. Multiply in the same way you would multiply two 2 -digit numbers. x 2 + 5 x – 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. Multiply: (x + 3)(x – 2) (x + 3) x____ (x – 2) 2 x – 6 x 2 + 3 x + 0 _____ x 2 + 5 x – 6 Here is another example. (x 2 – 3 x + 2)(x 2 – 3) (x 2 – 3 x + 2) x______ (x 2 – 3) – 3 x 2 + 9 x – 6 x 4 – 3 x 3 + 2 x 2 + 0 x + 0 _________ x 4 – 3 x 3 – 1 x 2 + 9 x – 6 Line up like terms.

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. Multiply: (x + 2)(x – 5) Each term in (x+2) is distributed to each term in (x – 5).

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