Multiplying

1
Multiplying
Polynomials
2

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

Objectives
3
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.
4
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.
5
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.
1
16
What weve 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.
6
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.
7
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.
7x2
x________
21x2
14x3y
Now multiply each term in the polynomial by the
monomial.
Keep track of negative signs.
8
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 2)
x________
Multiply in the same way you would multiply two
2-digit numbers.
6
2x
0
3x
x2
_________
6
5x
x2
Remember that we could use a vertical format when
multiplying a polynomial by monomial. We can do
the same here.
9
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 2)
x________
Line up like terms.
6
2x
(x2 3)
x____________
0
3x
x2
_________
6
9x
3x2
6
5x
x2
0
0x
2x2
3x3
x4
__________________
6
9x
1x2
3x3
x4
10
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 (x2) is distributed to each term in
(x 5).
Multiply (x 2)(x 5)
11
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.
12
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
13
You Try It!!
1. 2x2(3xy 7x 2y)
2. (x 4)(x 3)
3. (2y 3x)(y 2)
14
2x2(3xy 7x 2y)
2x2(3xy 7x 2y)
2x2(3xy) 2x2(7x) 2x2(2y)
Problem One
6x3y 14x2 4x2y
15
(x 4)(x 3)
(x 4)(x 3)
x(x) x(3) 4(x) 4(3)
Problem Two
x2 3x 4x 12
x2 x 12
16
(2y 3x)(y 2)
(2y 3x)(y 2)
2y(y) 2y(2) (3x)(y) (3x)(2)
Problem Three
2y2 4y 3xy 6x