View
283
Download
2
Category
Preview:
Citation preview
Essential Question
How do you multiply polynomials?
1. Multiply a polynomial by a monomial.
2. Multiply a polynomial by a polynomial.
The Distributive PropertyLook at the following expression:
3(x + 7) This expression is the sum of x and 7 multiplied by 3.
To simplify this expression we can distribute the multiplication by 3 to each number in the sum.
(3 • x) + (3 • 7)
3x + 21
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.
23x____7
7 • 3 = 21. Keep the 1 in the ones position then carry the 2 into the tens position.
1
2
7 • 2 = 14. Add the 2 from before and we get 16.
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) =
Then use the order of operations and the properties of exponents to simplify.
6x2y + 3xy2
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
7x2x________Align the terms vertically with the monomial under the polynomial.
Now multiply each term in the polynomial by the monomial.
– 21x214x3y
Keep track of negative signs.
To multiply a polynomial by another polynomial we use the distributive property as we did before.
Multiply: (x + 3)(x – 2)
Remember that we could use a vertical format when multiplying a polynomial by monomial. We can do the same here.
(x + 3)(x – 2)x________
Line up the terms by degree.
Multiply in the same way you would multiply two 2-digit numbers.
– 6 2x+ 0 + 3xx2_________– 6 + 5xx2
Multiply: (x + 3)(x – 2)
(x + 3)(x – 2)x________
– 6 2x+ 0 + 3xx2_________– 6 + 5xx2
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.(x2 – 3x + 2)(x2 – 3)
(x2 – 3x + 2)(x2 – 3)x____________
Line up like terms.
– 6 + 9x– 3x2
+ 0+ 0x+ 2x2– 3x3x4__________________– 6 + 9x – 1x2 – 3x3 x4
It is also advantageous to multiply polynomials without rewriting them in a vertical format.
Multiply: (x + 2)(x – 5)
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).
(x + 2)(x – 5)
This pattern for multiplying polynomials is called FOIL.
Multiply the First terms.
Multiply the Outside terms.
Multiply the Inside terms.
Multiply the Last terms.
F
O
I
L After you multiply, collect like terms.
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) + 2x2(7x) + 2x2(–2y)
2x2(3xy + 7x – 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
Recommended