Multiplying Polynomials
-Distributive property
-FOIL
-Box Method
To multiply a polynomial by a monomial, use the distributive
property and the rule for multiplying exponential expressions.
Examples: 1. Multiply: 2x(3x2 + 2x – 1).
= 2x(3x2 ) + 2x(2x) + 2x(–1)
= 6x3 + 4x2 – 2x
2. Multiply: – 3x2y(5x2 – 2xy + 7y2).
= – 3x2y(5x2 ) – 3x2y(–2xy) – 3x2y(7y2)
= – 15x4y + 6x3y2 – 21x2y3
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Multiplying Polynomials-Try it!
Multiply: – 2xy(8x2 +2xy – 5y2).
= – 2xy(8x2 ) – 2xy( 2xy) – 2xy(-5y2)
= – 16x3y – 6x2y2 + 10xy3
3
Try it!.
Multiply the polynomial by the monomial.
1) 3(x + 4)
2)
3x 12
2a(a  5)
Distributive Property
2a  10a
2
6k(2k  4k  3)
2
3)
12k  24k  18k
3
2
To multiply two binomials use a method called FOIL,
which is based on the distributive property. The letters
of FOIL stand for First, Outer, Inner, and Last.
1. Multiply the first terms.
2. Multiply the outer terms.
3. Multiply the inner terms.
4. Multiply the last terms.
5. Combine like terms.
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5
For use with the product of binomials only!
(x  3)(x  5)
First
x
2
Outer
Inner
Last
For use with the product of binomials only!
(x  3)(x  5)
First
x
2
Outer
5x
Inner
Last
For use with the product of binomials only!
(x  3)(x  5)
First
x
2
Outer
5x
Inner
3x
Last
For use with the product of binomials only!
(x  3)(x  5)
First
x
2
Outer
5x
Inner
Last
3x
15
For use with the product of binomials only!
(x  3)(x  5)
First
x
2
Outer
5x
Inner
Last
3x
15
x  2x  15
2
Examples: 1. Multiply: (2x + 1)(7x – 5).
First
Outer
Inner
Last
= 2x(7x) + 2x(–5) + (1)(7x) + (1)(–5)
= 14x2 – 10x + 7x – 5
= 14x2 – 3x – 5
2. Multiply: (5x – 3y)(7x + 6y).
First
Outer
Inner
Last
= 5x(7x) + 5x(6y) + (– 3y)(7x) + (– 3y)(6y)
= 35x2 + 30xy – 21yx – 18y2
= 35x2 + 9xy – 18y2
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11
Try...
(m  3)(m  6)
First
m
2
Outer
Inner
Last
6m
3m
18
m  9m  18
2
The third method is the Box Method.
This method works for every problem!
Here’s how you do it.
Multiply (3x – 5)(5x + 2)
Draw a box. Write a
polynomial on the top and
side of a box. It does not
matter which goes where.
This will be modeled in the
next problem along with
FOIL.
3x
5x
+2
-5
Multiply (3x - 5)(5x + 2)
First terms: 15x2
Outer terms: +6x
Inner terms: -25x
Last terms: -10
Combine like terms.
15x2 - 19x – 10
3x
5x
-5
15x2 -25x
+2 +6x
-10
You have 3 techniques. Pick the one you like the best!
Try it! Multiply (7p - 2)(3p - 4)
First terms: 21p2
Outer terms: -28p
Inner terms: -6p
Last terms: +8
Combine like terms.
21p2 – 34p + 8
7p
3p
-2
21p2 -6p
-4 -28p
+8
To multiply two polynomials, apply the distributive property.
Example: Multiply: (x – 1)(2x2 + 7x + 3).
= (x – 1)(2x2) + (x – 1)(7x) + (x – 1)(3)
= 2x3 – 2x2 + 7x2 – 7x + 3x – 3
= 2x3 + 5x2 – 4x – 3
Two polynomials can also be multiplied using a vertical format.
Example:
2x2 + 7x + 3
x–1
– 2x2 – 7x – 3
2x3 + 7x2 + 3x
2x3 + 5x2 – 4x – 3x
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Multiply – 1(2x2 + 7x + 3).
Multiply x(2x2 + 7x + 3).
Add the terms in each column.
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Multiply (2x - 5)(x2 - 5x + 4)
You cannot use FOIL because they are not
BOTH binomials. You must use the
distributive property.
2x(x2 - 5x + 4) - 5(x2 - 5x + 4)
2x3 - 10x2 + 8x - 5x2 + 25x - 20
Group and combine like terms.
2x3 - 10x2 - 5x2 + 8x + 25x - 20
2x3 - 15x2 + 33x - 20
Multiply (2x - 5)(x2 - 5x + 4)
You cannot use FOIL because they are not BOTH
binomials. You must use the distributive property or
box method.
2x
-5
x2
-5x
+4
2x3
-10x2
+8x
-5x2 +25x
-20
Almost
done!
Go to
the next
slide!
Multiply (2x - 5)(x2 - 5x + 4)
Combine like terms!
x2
-5x
+4
2x
2x3
-10x2
+8x
-5
-5x2 +25x
-20
2x3 – 15x2 + 33x - 20
Example: The length of a rectangle is (x + 5) ft. The width
is (x – 6) ft. Find the area of the rectangle in terms of
the variable x.
x–6
A = L · W = Area
L = (x + 5) ft
W = (x – 6) ft
x+5
A = (x + 5)(x – 6 ) = x2 – 6x + 5x – 30
= x2 – x – 30
The area is (x2 – x – 30) ft2.
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