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8/10/2019 9 5 Multiplying Polynomials
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1. Multiply a polynomial by a monomial.
2. Multiply a polynomial by a polynomial.
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The Distributive Property
Look 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
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Whenever we multiply two numbers, we are putting the distributive
property to work.
7(23) We can rewrite 23 as (20 + 3) then theproblem 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.
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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 weve really done in the second
step, is multiply 7 by 20, then add the20 left over from the first step to get
160. We add this to the 1 to get 161.
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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
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We can also multiply a polynomial and a monomial using a vertical
format in the same way we would multiply two numbers.
Multiply: 7x2(2xy3x2)
2xy3x2
7x2x________
Align the terms vertically with the
monomial under the polynomial.
Now multiply each term in thepolynomial by the monomial.
21x2
14x3
y
Keep track of negative
signs.
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To multiply a polynomial by another polynomial we use the
distributive property as we did before.
Multiply: (x + 3)(x2)
Remember that we could use a vertical format when multiplying a
polynomial by monomial. We can do the same here.
(x + 3)(x2)x________
Line up the terms by degree.
Multiply in the same way
you would multiply two 2-
digit numbers.
62x+ 0+ 3xx2_________
6+ 5xx2
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Multiply: (x + 3)(x2)(x + 3)(x2)x________
62x+ 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.
(x23x + 2)(x23)
(x23x + 2)
(x2 3)x____________Line up like terms.
6+ 9x3x2
+ 0+ 0x+ 2x23x3x4__________________6+ 9x1x23x3x4
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It is also advantageous to multiply polynomials without rewriting
them in a vertical format.
Multiply: (x + 2)(x5)
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 (x5).
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(x + 2)(x5)
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.
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Example: (x6)(2x + 1)
x(2x) + x(1)(6)2x 6(1)
2x2+ x12x6
2x211x6
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2x2(3xy + 7x2y)
2x2(3xy) + 2x2(7x) + 2x2(2y)
2x2(3xy + 7x2y)
6x3y + 14x24x2y
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(x + 4)(x3)
(x + 4)(x3)
x(x) + x(3) + 4(x) + 4(3)
x23x + 4x12
x2+ x12
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(2y3x)(y2)
(2y3x)(y2)
2y(y) + 2y(2) + (3x)(y) + (3x)(2)
2y24y3xy + 6x