2Objectives Multiply a polynomial by a monomial. Multiply a polynomial by a polynomial.Objectives
3The Distributive Property ReviewThe Distributive PropertyLook at the following expression:3(x + 7)This expression is the sum of x and 7 multiplied by 3.(3 • x)+(3 • 7)3x + 21To simplify this expression we can distribute the multiplication by 3 to each number in the sum.
4ReviewWhenever 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) = = 161When we learn to multiply multi-digit numbers, we do the same thing in a vertical format.
5Review7 • 3 = 21. Keep the 1 in the ones position then carry the 2 into the tens position.223x____77 • 2 = 14. Add the 2 from before and we get 16.161What 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 We add this to the 1 to get 161.
6Multiplying a Polynomial by a MonomialMultiply: 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 + 3xy2Then use the order of operations and the properties of exponents to simplify.
7Multiplying a Polynomial by a MonomialWe 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 – 3x2Align the terms vertically with the monomial under the polynomial.x________7x214x3y– 21x4Now multiply each term in the polynomial by the monomial.Keep track of negative signs.
8Multiplying a Polynomial by a PolynomialTo 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+ 0x2+ 5x– 6Remember that we could use a vertical format when multiplying a polynomial by monomial. We can do the same here.
9Multiplying a Polynomial by a PolynomialTo 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– 6x____________(x – 3)x2_________+ 3x+ 0– 3x2+ 9x– 6x2+ 5x– 6x4__________________– 3x3+ 2x2+ 0x+ 0x4– 3x3– 1x2+ 9x– 6
10Multiplying a Polynomial by a PolynomialIt 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)
11Multiplying a Polynomial by a PolynomialMultiply the First terms.OMultiply the Outside terms.F(x + 2)(x – 5)Multiply the Inside terms.Multiply the Last terms.IAfter you multiply, collect like terms.LThis pattern for multiplying polynomials is called FOIL.
12Multiplying a Polynomial by a PolynomialExample:(x – 6)(2x + 1)x(2x)+ x(1)– (6)2x– 6(1)2x2 + x – 12x – 62x2 – 11x – 6