Download Multiply a polynomial

1. Multiply a polynomial by a monomial.
2. Multiply a polynomial by a polynomial.
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.
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
161
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.
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.
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
7x2
x________
14x3y – 21x2
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.
To multiply a polynomial by another polynomial we use the
distributive property as we did before.
Multiply: (x + 3)(x – 2)
(x + 3)
(x – 2)
x________
2x – 6
x2 + 3x + 0
_________
x2 + 5x – 6
Line up the terms by degree.
Multiply in the same way
you would multiply two 2digit numbers.
Remember that we could use a vertical format when multiplying a
polynomial by monomial. We can do the same here.
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.
2
2x – 6
(x
– 3)
x____________
x2 + 3x + 0
_________
– 3x2 + 9x – 6
x2 + 5x – 6
x4 – 3x3 + 2x2 + 0x + 0
__________________
x4 – 3x3 – 1x2 + 9x – 6
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).
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.
Example:
(x – 6)(2x + 1)
x(2x) + x(1) – (6)2x – 6(1)
2x2 + x – 12x – 6
2x2 – 11x – 6
1. 2x2(3xy + 7x – 2y)
2. (x + 4)(x – 3)
3. (2y – 3x)(y – 2)
2x2(3xy + 7x – 2y)
2x2(3xy + 7x – 2y)
2x2(3xy) + 2x2(7x) + 2x2(–2y)
6x3y + 14x2 – 4x2y
(x + 4)(x – 3)
(x + 4)(x – 3)
x(x) + x(–3) + 4(x) + 4(–3)
x2 – 3x + 4x – 12
x2 + x – 12
(2y – 3x)(y – 2)
(2y – 3x)(y – 2)
2y(y) + 2y(–2) + (–3x)(y) + (–3x)(–2)
2y2 – 4y – 3xy + 6x